Rational Exponents - Ninth Grade Math
Introduction to Rational Exponents
Rational Exponent: An exponent that is expressed as a fraction (ratio of two integers)
Also Called: Fractional exponents
General Form: $a^{\frac{m}{n}}$ where $a$ is the base, $m$ is the numerator, $n$ is the denominator
Key Idea: Rational exponents represent both powers and roots combined
Also Called: Fractional exponents
General Form: $a^{\frac{m}{n}}$ where $a$ is the base, $m$ is the numerator, $n$ is the denominator
Key Idea: Rational exponents represent both powers and roots combined
Fundamental Rational Exponent Formulas:
1. Basic Form:
$$a^{\frac{1}{n}} = \sqrt[n]{a}$$
The denominator ($n$) represents the root
2. General Form:
$$a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$$
• Numerator ($m$) = power
• Denominator ($n$) = root
Examples:
• $a^{\frac{1}{2}} = \sqrt{a}$ (square root)
• $a^{\frac{1}{3}} = \sqrt[3]{a}$ (cube root)
• $a^{\frac{2}{3}} = \sqrt[3]{a^2}$ or $\left(\sqrt[3]{a}\right)^2$
1. Basic Form:
$$a^{\frac{1}{n}} = \sqrt[n]{a}$$
The denominator ($n$) represents the root
2. General Form:
$$a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$$
• Numerator ($m$) = power
• Denominator ($n$) = root
Examples:
• $a^{\frac{1}{2}} = \sqrt{a}$ (square root)
• $a^{\frac{1}{3}} = \sqrt[3]{a}$ (cube root)
• $a^{\frac{2}{3}} = \sqrt[3]{a^2}$ or $\left(\sqrt[3]{a}\right)^2$
Understanding Rational Exponents:
• The numerator tells you the power to raise the base to
• The denominator tells you which root to take
• You can take the root first, then the power OR power first, then the root
• Both methods give the same result!
Example: $8^{\frac{2}{3}}$
• Method 1: $\sqrt[3]{8^2} = \sqrt[3]{64} = 4$
• Method 2: $\left(\sqrt[3]{8}\right)^2 = 2^2 = 4$ ✓
• The numerator tells you the power to raise the base to
• The denominator tells you which root to take
• You can take the root first, then the power OR power first, then the root
• Both methods give the same result!
Example: $8^{\frac{2}{3}}$
• Method 1: $\sqrt[3]{8^2} = \sqrt[3]{64} = 4$
• Method 2: $\left(\sqrt[3]{8}\right)^2 = 2^2 = 4$ ✓
1-2. Evaluate Integers Raised to Rational Exponents
1. Positive Rational Exponents
Evaluating: Finding the numerical value of an expression with rational exponents
Steps to Evaluate $a^{\frac{m}{n}}$:
Method 1 (Root First):
Step 1: Find the $n$th root of $a$: $\sqrt[n]{a}$
Step 2: Raise result to the power $m$
Method 2 (Power First):
Step 1: Raise $a$ to the power $m$: $a^m$
Step 2: Find the $n$th root of result: $\sqrt[n]{a^m}$
Tip: Method 1 usually involves smaller numbers!
Method 1 (Root First):
Step 1: Find the $n$th root of $a$: $\sqrt[n]{a}$
Step 2: Raise result to the power $m$
Method 2 (Power First):
Step 1: Raise $a$ to the power $m$: $a^m$
Step 2: Find the $n$th root of result: $\sqrt[n]{a^m}$
Tip: Method 1 usually involves smaller numbers!
Example 1: Evaluate $16^{\frac{1}{2}}$
This asks for the square root of 16
$16^{\frac{1}{2}} = \sqrt{16} = 4$
Answer: 4
This asks for the square root of 16
$16^{\frac{1}{2}} = \sqrt{16} = 4$
Answer: 4
Example 2: Evaluate $27^{\frac{1}{3}}$
This asks for the cube root of 27
$27^{\frac{1}{3}} = \sqrt[3]{27} = 3$
Answer: 3
This asks for the cube root of 27
$27^{\frac{1}{3}} = \sqrt[3]{27} = 3$
Answer: 3
Example 3: Evaluate $8^{\frac{2}{3}}$
Method 1 (Root first - easier!):
$8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$
Method 2 (Power first):
$8^{\frac{2}{3}} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4$
Answer: 4
Method 1 (Root first - easier!):
$8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$
Method 2 (Power first):
$8^{\frac{2}{3}} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4$
Answer: 4
Example 4: Evaluate $16^{\frac{3}{4}}$
Root first:
$16^{\frac{3}{4}} = \left(\sqrt[4]{16}\right)^3 = 2^3 = 8$
Answer: 8
Root first:
$16^{\frac{3}{4}} = \left(\sqrt[4]{16}\right)^3 = 2^3 = 8$
Answer: 8
Example 5: Evaluate $32^{\frac{2}{5}}$
$32^{\frac{2}{5}} = \left(\sqrt[5]{32}\right)^2 = 2^2 = 4$
Answer: 4
$32^{\frac{2}{5}} = \left(\sqrt[5]{32}\right)^2 = 2^2 = 4$
Answer: 4
Example 6: Evaluate $125^{\frac{2}{3}}$
$125^{\frac{2}{3}} = \left(\sqrt[3]{125}\right)^2 = 5^2 = 25$
Answer: 25
$125^{\frac{2}{3}} = \left(\sqrt[3]{125}\right)^2 = 5^2 = 25$
Answer: 25
Example 7: Evaluate $81^{\frac{3}{4}}$
$81^{\frac{3}{4}} = \left(\sqrt[4]{81}\right)^3 = 3^3 = 27$
Answer: 27
$81^{\frac{3}{4}} = \left(\sqrt[4]{81}\right)^3 = 3^3 = 27$
Answer: 27
2. Negative Rational Exponents
Negative Rational Exponent Rule:
$$a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}} = \frac{1}{\sqrt[n]{a^m}}$$
Just like integer exponents, negative means "take the reciprocal"
$$a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}} = \frac{1}{\sqrt[n]{a^m}}$$
Just like integer exponents, negative means "take the reciprocal"
Example 1: Evaluate $16^{-\frac{1}{2}}$
$16^{-\frac{1}{2}} = \frac{1}{16^{\frac{1}{2}}} = \frac{1}{\sqrt{16}} = \frac{1}{4}$
Answer: $\frac{1}{4}$
$16^{-\frac{1}{2}} = \frac{1}{16^{\frac{1}{2}}} = \frac{1}{\sqrt{16}} = \frac{1}{4}$
Answer: $\frac{1}{4}$
Example 2: Evaluate $8^{-\frac{2}{3}}$
$8^{-\frac{2}{3}} = \frac{1}{8^{\frac{2}{3}}} = \frac{1}{\left(\sqrt[3]{8}\right)^2} = \frac{1}{2^2} = \frac{1}{4}$
Answer: $\frac{1}{4}$
$8^{-\frac{2}{3}} = \frac{1}{8^{\frac{2}{3}}} = \frac{1}{\left(\sqrt[3]{8}\right)^2} = \frac{1}{2^2} = \frac{1}{4}$
Answer: $\frac{1}{4}$
Example 3: Evaluate $27^{-\frac{2}{3}}$
$27^{-\frac{2}{3}} = \frac{1}{27^{\frac{2}{3}}} = \frac{1}{\left(\sqrt[3]{27}\right)^2} = \frac{1}{3^2} = \frac{1}{9}$
Answer: $\frac{1}{9}$
$27^{-\frac{2}{3}} = \frac{1}{27^{\frac{2}{3}}} = \frac{1}{\left(\sqrt[3]{27}\right)^2} = \frac{1}{3^2} = \frac{1}{9}$
Answer: $\frac{1}{9}$
Example 4: Evaluate $32^{-\frac{3}{5}}$
$32^{-\frac{3}{5}} = \frac{1}{32^{\frac{3}{5}}} = \frac{1}{\left(\sqrt[5]{32}\right)^3} = \frac{1}{2^3} = \frac{1}{8}$
Answer: $\frac{1}{8}$
$32^{-\frac{3}{5}} = \frac{1}{32^{\frac{3}{5}}} = \frac{1}{\left(\sqrt[5]{32}\right)^3} = \frac{1}{2^3} = \frac{1}{8}$
Answer: $\frac{1}{8}$
Quick Recognition:
Recognize perfect powers to make evaluation easier:
• $4 = 2^2$, $8 = 2^3$, $16 = 2^4$, $32 = 2^5$, $64 = 2^6$
• $9 = 3^2$, $27 = 3^3$, $81 = 3^4$
• $25 = 5^2$, $125 = 5^3$
• $36 = 6^2$, $49 = 7^2$, $100 = 10^2$
Recognize perfect powers to make evaluation easier:
• $4 = 2^2$, $8 = 2^3$, $16 = 2^4$, $32 = 2^5$, $64 = 2^6$
• $9 = 3^2$, $27 = 3^3$, $81 = 3^4$
• $25 = 5^2$, $125 = 5^3$
• $36 = 6^2$, $49 = 7^2$, $100 = 10^2$
3. Multiplication with Rational Exponents
Product Rule: When multiplying powers with the same base, add the exponents
Multiplication Rule for Rational Exponents:
$$a^{\frac{m}{n}} \times a^{\frac{p}{q}} = a^{\frac{m}{n} + \frac{p}{q}}$$
Important: Bases must be the SAME!
Additional Rule:
$$a^{\frac{m}{n}} \times b^{\frac{m}{n}} = (ab)^{\frac{m}{n}}$$
When exponents are the same, multiply bases
$$a^{\frac{m}{n}} \times a^{\frac{p}{q}} = a^{\frac{m}{n} + \frac{p}{q}}$$
Important: Bases must be the SAME!
Additional Rule:
$$a^{\frac{m}{n}} \times b^{\frac{m}{n}} = (ab)^{\frac{m}{n}}$$
When exponents are the same, multiply bases
Example 1: Simplify $x^{\frac{1}{2}} \times x^{\frac{1}{3}}$
Add exponents (find common denominator):
$x^{\frac{1}{2}} \times x^{\frac{1}{3}} = x^{\frac{3}{6} + \frac{2}{6}} = x^{\frac{5}{6}}$
Answer: $x^{\frac{5}{6}}$
Add exponents (find common denominator):
$x^{\frac{1}{2}} \times x^{\frac{1}{3}} = x^{\frac{3}{6} + \frac{2}{6}} = x^{\frac{5}{6}}$
Answer: $x^{\frac{5}{6}}$
Example 2: Simplify $5^{\frac{2}{3}} \times 5^{\frac{1}{3}}$
$5^{\frac{2}{3}} \times 5^{\frac{1}{3}} = 5^{\frac{2}{3} + \frac{1}{3}} = 5^{\frac{3}{3}} = 5^1 = 5$
Answer: 5
$5^{\frac{2}{3}} \times 5^{\frac{1}{3}} = 5^{\frac{2}{3} + \frac{1}{3}} = 5^{\frac{3}{3}} = 5^1 = 5$
Answer: 5
Example 3: Simplify $x^{\frac{3}{4}} \times x^{\frac{1}{2}}$
$x^{\frac{3}{4}} \times x^{\frac{1}{2}} = x^{\frac{3}{4} + \frac{2}{4}} = x^{\frac{5}{4}}$
Answer: $x^{\frac{5}{4}}$
$x^{\frac{3}{4}} \times x^{\frac{1}{2}} = x^{\frac{3}{4} + \frac{2}{4}} = x^{\frac{5}{4}}$
Answer: $x^{\frac{5}{4}}$
Example 4: Simplify $2^{\frac{1}{3}} \times 8^{\frac{1}{3}}$
Method 1: Combine bases (same exponent):
$2^{\frac{1}{3}} \times 8^{\frac{1}{3}} = (2 \times 8)^{\frac{1}{3}} = 16^{\frac{1}{3}} = \sqrt[3]{16}$
Method 2: Express 8 as power of 2:
$2^{\frac{1}{3}} \times (2^3)^{\frac{1}{3}} = 2^{\frac{1}{3}} \times 2^1 = 2^{\frac{1}{3} + 1} = 2^{\frac{4}{3}}$
Answer: $2^{\frac{4}{3}}$ or $\sqrt[3]{16}$
Method 1: Combine bases (same exponent):
$2^{\frac{1}{3}} \times 8^{\frac{1}{3}} = (2 \times 8)^{\frac{1}{3}} = 16^{\frac{1}{3}} = \sqrt[3]{16}$
Method 2: Express 8 as power of 2:
$2^{\frac{1}{3}} \times (2^3)^{\frac{1}{3}} = 2^{\frac{1}{3}} \times 2^1 = 2^{\frac{1}{3} + 1} = 2^{\frac{4}{3}}$
Answer: $2^{\frac{4}{3}}$ or $\sqrt[3]{16}$
Example 5: Simplify $a^{\frac{2}{5}} \times a^{\frac{3}{5}} \times a^{\frac{1}{5}}$
$a^{\frac{2}{5}} \times a^{\frac{3}{5}} \times a^{\frac{1}{5}} = a^{\frac{2+3+1}{5}} = a^{\frac{6}{5}}$
Answer: $a^{\frac{6}{5}}$
$a^{\frac{2}{5}} \times a^{\frac{3}{5}} \times a^{\frac{1}{5}} = a^{\frac{2+3+1}{5}} = a^{\frac{6}{5}}$
Answer: $a^{\frac{6}{5}}$
4. Division with Rational Exponents
Quotient Rule: When dividing powers with the same base, subtract the exponents
Division Rule for Rational Exponents:
$$\frac{a^{\frac{m}{n}}}{a^{\frac{p}{q}}} = a^{\frac{m}{n} - \frac{p}{q}}$$
Additional Rule:
$$\frac{a^{\frac{m}{n}}}{b^{\frac{m}{n}}} = \left(\frac{a}{b}\right)^{\frac{m}{n}}$$
When exponents are the same, divide bases
$$\frac{a^{\frac{m}{n}}}{a^{\frac{p}{q}}} = a^{\frac{m}{n} - \frac{p}{q}}$$
Additional Rule:
$$\frac{a^{\frac{m}{n}}}{b^{\frac{m}{n}}} = \left(\frac{a}{b}\right)^{\frac{m}{n}}$$
When exponents are the same, divide bases
Example 1: Simplify $\frac{x^{\frac{5}{6}}}{x^{\frac{1}{6}}}$
$\frac{x^{\frac{5}{6}}}{x^{\frac{1}{6}}} = x^{\frac{5}{6} - \frac{1}{6}} = x^{\frac{4}{6}} = x^{\frac{2}{3}}$
Answer: $x^{\frac{2}{3}}$
$\frac{x^{\frac{5}{6}}}{x^{\frac{1}{6}}} = x^{\frac{5}{6} - \frac{1}{6}} = x^{\frac{4}{6}} = x^{\frac{2}{3}}$
Answer: $x^{\frac{2}{3}}$
Example 2: Simplify $\frac{7^{\frac{3}{4}}}{7^{\frac{1}{4}}}$
$\frac{7^{\frac{3}{4}}}{7^{\frac{1}{4}}} = 7^{\frac{3}{4} - \frac{1}{4}} = 7^{\frac{2}{4}} = 7^{\frac{1}{2}} = \sqrt{7}$
Answer: $7^{\frac{1}{2}}$ or $\sqrt{7}$
$\frac{7^{\frac{3}{4}}}{7^{\frac{1}{4}}} = 7^{\frac{3}{4} - \frac{1}{4}} = 7^{\frac{2}{4}} = 7^{\frac{1}{2}} = \sqrt{7}$
Answer: $7^{\frac{1}{2}}$ or $\sqrt{7}$
Example 3: Simplify $\frac{y^{\frac{7}{5}}}{y^{\frac{2}{5}}}$
$\frac{y^{\frac{7}{5}}}{y^{\frac{2}{5}}} = y^{\frac{7}{5} - \frac{2}{5}} = y^{\frac{5}{5}} = y^1 = y$
Answer: $y$
$\frac{y^{\frac{7}{5}}}{y^{\frac{2}{5}}} = y^{\frac{7}{5} - \frac{2}{5}} = y^{\frac{5}{5}} = y^1 = y$
Answer: $y$
Example 4: Simplify $\frac{a^{\frac{2}{3}}}{a^{\frac{5}{6}}}$
Find common denominator:
$\frac{a^{\frac{2}{3}}}{a^{\frac{5}{6}}} = a^{\frac{4}{6} - \frac{5}{6}} = a^{-\frac{1}{6}} = \frac{1}{a^{\frac{1}{6}}}$
Answer: $a^{-\frac{1}{6}}$ or $\frac{1}{a^{\frac{1}{6}}}$
Find common denominator:
$\frac{a^{\frac{2}{3}}}{a^{\frac{5}{6}}} = a^{\frac{4}{6} - \frac{5}{6}} = a^{-\frac{1}{6}} = \frac{1}{a^{\frac{1}{6}}}$
Answer: $a^{-\frac{1}{6}}$ or $\frac{1}{a^{\frac{1}{6}}}$
Example 5: Simplify $\frac{16^{\frac{3}{4}}}{2^{\frac{3}{4}}}$
Same exponent, divide bases:
$\frac{16^{\frac{3}{4}}}{2^{\frac{3}{4}}} = \left(\frac{16}{2}\right)^{\frac{3}{4}} = 8^{\frac{3}{4}} = \left(\sqrt[4]{8}\right)^3$
Or simplify: $8 = 2^3$, so $8^{\frac{3}{4}} = (2^3)^{\frac{3}{4}} = 2^{\frac{9}{4}}$
Answer: $8^{\frac{3}{4}}$ or $2^{\frac{9}{4}}$
Same exponent, divide bases:
$\frac{16^{\frac{3}{4}}}{2^{\frac{3}{4}}} = \left(\frac{16}{2}\right)^{\frac{3}{4}} = 8^{\frac{3}{4}} = \left(\sqrt[4]{8}\right)^3$
Or simplify: $8 = 2^3$, so $8^{\frac{3}{4}} = (2^3)^{\frac{3}{4}} = 2^{\frac{9}{4}}$
Answer: $8^{\frac{3}{4}}$ or $2^{\frac{9}{4}}$
5. Power Rule with Rational Exponents
Power Rule: When raising a power to another power, multiply the exponents
Power Rule for Rational Exponents:
$$\left(a^{\frac{m}{n}}\right)^{\frac{p}{q}} = a^{\frac{m}{n} \times \frac{p}{q}} = a^{\frac{mp}{nq}}$$
Additional Rules:
$$(ab)^{\frac{m}{n}} = a^{\frac{m}{n}} \times b^{\frac{m}{n}}$$ (Power of a product)
$$\left(\frac{a}{b}\right)^{\frac{m}{n}} = \frac{a^{\frac{m}{n}}}{b^{\frac{m}{n}}}$$ (Power of a quotient)
$$\left(a^{\frac{m}{n}}\right)^{\frac{p}{q}} = a^{\frac{m}{n} \times \frac{p}{q}} = a^{\frac{mp}{nq}}$$
Additional Rules:
$$(ab)^{\frac{m}{n}} = a^{\frac{m}{n}} \times b^{\frac{m}{n}}$$ (Power of a product)
$$\left(\frac{a}{b}\right)^{\frac{m}{n}} = \frac{a^{\frac{m}{n}}}{b^{\frac{m}{n}}}$$ (Power of a quotient)
Example 1: Simplify $\left(x^{\frac{2}{3}}\right)^{\frac{3}{4}}$
$\left(x^{\frac{2}{3}}\right)^{\frac{3}{4}} = x^{\frac{2}{3} \times \frac{3}{4}} = x^{\frac{6}{12}} = x^{\frac{1}{2}}$
Answer: $x^{\frac{1}{2}}$
$\left(x^{\frac{2}{3}}\right)^{\frac{3}{4}} = x^{\frac{2}{3} \times \frac{3}{4}} = x^{\frac{6}{12}} = x^{\frac{1}{2}}$
Answer: $x^{\frac{1}{2}}$
Example 2: Simplify $\left(4^{\frac{1}{2}}\right)^3$
$\left(4^{\frac{1}{2}}\right)^3 = 4^{\frac{1}{2} \times 3} = 4^{\frac{3}{2}} = \left(\sqrt{4}\right)^3 = 2^3 = 8$
Answer: $4^{\frac{3}{2}}$ or $8$
$\left(4^{\frac{1}{2}}\right)^3 = 4^{\frac{1}{2} \times 3} = 4^{\frac{3}{2}} = \left(\sqrt{4}\right)^3 = 2^3 = 8$
Answer: $4^{\frac{3}{2}}$ or $8$
Example 3: Simplify $\left(y^{\frac{3}{5}}\right)^{\frac{5}{3}}$
$\left(y^{\frac{3}{5}}\right)^{\frac{5}{3}} = y^{\frac{3}{5} \times \frac{5}{3}} = y^{\frac{15}{15}} = y^1 = y$
Answer: $y$
$\left(y^{\frac{3}{5}}\right)^{\frac{5}{3}} = y^{\frac{3}{5} \times \frac{5}{3}} = y^{\frac{15}{15}} = y^1 = y$
Answer: $y$
Example 4: Simplify $(8x^3)^{\frac{2}{3}}$
$(8x^3)^{\frac{2}{3}} = 8^{\frac{2}{3}} \times (x^3)^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2 \times x^{3 \times \frac{2}{3}} = 2^2 \times x^2 = 4x^2$
Answer: $4x^2$
$(8x^3)^{\frac{2}{3}} = 8^{\frac{2}{3}} \times (x^3)^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2 \times x^{3 \times \frac{2}{3}} = 2^2 \times x^2 = 4x^2$
Answer: $4x^2$
Example 5: Simplify $\left(\frac{x^4}{y^2}\right)^{\frac{1}{2}}$
$\left(\frac{x^4}{y^2}\right)^{\frac{1}{2}} = \frac{(x^4)^{\frac{1}{2}}}{(y^2)^{\frac{1}{2}}} = \frac{x^{4 \times \frac{1}{2}}}{y^{2 \times \frac{1}{2}}} = \frac{x^2}{y^1} = \frac{x^2}{y}$
Answer: $\frac{x^2}{y}$
$\left(\frac{x^4}{y^2}\right)^{\frac{1}{2}} = \frac{(x^4)^{\frac{1}{2}}}{(y^2)^{\frac{1}{2}}} = \frac{x^{4 \times \frac{1}{2}}}{y^{2 \times \frac{1}{2}}} = \frac{x^2}{y^1} = \frac{x^2}{y}$
Answer: $\frac{x^2}{y}$
Example 6: Simplify $\left(27x^6\right)^{\frac{1}{3}}$
$\left(27x^6\right)^{\frac{1}{3}} = 27^{\frac{1}{3}} \times (x^6)^{\frac{1}{3}} = \sqrt[3]{27} \times x^{6 \times \frac{1}{3}} = 3 \times x^2 = 3x^2$
Answer: $3x^2$
$\left(27x^6\right)^{\frac{1}{3}} = 27^{\frac{1}{3}} \times (x^6)^{\frac{1}{3}} = \sqrt[3]{27} \times x^{6 \times \frac{1}{3}} = 3 \times x^2 = 3x^2$
Answer: $3x^2$
6. Simplify Expressions Involving Rational Exponents
Simplifying: Using all exponent rules to write expressions in simplest form with positive exponents
Strategy for Simplifying:
Step 1: Apply power rule to parentheses first
Step 2: Use product rule for multiplication (add exponents)
Step 3: Use quotient rule for division (subtract exponents)
Step 4: Simplify fractions in exponents
Step 5: Convert negative exponents to positive
Step 6: Simplify coefficients if possible
Step 1: Apply power rule to parentheses first
Step 2: Use product rule for multiplication (add exponents)
Step 3: Use quotient rule for division (subtract exponents)
Step 4: Simplify fractions in exponents
Step 5: Convert negative exponents to positive
Step 6: Simplify coefficients if possible
Example 1: Simplify $\frac{x^{\frac{3}{4}} \times x^{\frac{1}{4}}}{x^{\frac{1}{2}}}$
Step 1: Multiply numerator
$\frac{x^{\frac{3}{4} + \frac{1}{4}}}{x^{\frac{1}{2}}} = \frac{x^{\frac{4}{4}}}{x^{\frac{1}{2}}} = \frac{x^1}{x^{\frac{1}{2}}}$
Step 2: Divide
$\frac{x}{x^{\frac{1}{2}}} = x^{1 - \frac{1}{2}} = x^{\frac{1}{2}}$
Answer: $x^{\frac{1}{2}}$
Step 1: Multiply numerator
$\frac{x^{\frac{3}{4} + \frac{1}{4}}}{x^{\frac{1}{2}}} = \frac{x^{\frac{4}{4}}}{x^{\frac{1}{2}}} = \frac{x^1}{x^{\frac{1}{2}}}$
Step 2: Divide
$\frac{x}{x^{\frac{1}{2}}} = x^{1 - \frac{1}{2}} = x^{\frac{1}{2}}$
Answer: $x^{\frac{1}{2}}$
Example 2: Simplify $\left(a^{\frac{1}{3}} \times a^{\frac{2}{3}}\right)^2$
$\left(a^{\frac{1}{3} + \frac{2}{3}}\right)^2 = \left(a^{\frac{3}{3}}\right)^2 = (a^1)^2 = a^2$
Answer: $a^2$
$\left(a^{\frac{1}{3} + \frac{2}{3}}\right)^2 = \left(a^{\frac{3}{3}}\right)^2 = (a^1)^2 = a^2$
Answer: $a^2$
Example 3: Simplify $\frac{(x^2)^{\frac{3}{4}}}{x^{\frac{1}{2}}}$
$\frac{x^{2 \times \frac{3}{4}}}{x^{\frac{1}{2}}} = \frac{x^{\frac{6}{4}}}{x^{\frac{1}{2}}} = \frac{x^{\frac{3}{2}}}{x^{\frac{1}{2}}} = x^{\frac{3}{2} - \frac{1}{2}} = x^1 = x$
Answer: $x$
$\frac{x^{2 \times \frac{3}{4}}}{x^{\frac{1}{2}}} = \frac{x^{\frac{6}{4}}}{x^{\frac{1}{2}}} = \frac{x^{\frac{3}{2}}}{x^{\frac{1}{2}}} = x^{\frac{3}{2} - \frac{1}{2}} = x^1 = x$
Answer: $x$
Example 4: Simplify $\left(\frac{16x^4}{y^8}\right)^{\frac{3}{4}}$
$\left(\frac{16x^4}{y^8}\right)^{\frac{3}{4}} = \frac{16^{\frac{3}{4}} \times (x^4)^{\frac{3}{4}}}{(y^8)^{\frac{3}{4}}}$
$= \frac{(2^4)^{\frac{3}{4}} \times x^3}{y^6} = \frac{2^3 \times x^3}{y^6} = \frac{8x^3}{y^6}$
Answer: $\frac{8x^3}{y^6}$
$\left(\frac{16x^4}{y^8}\right)^{\frac{3}{4}} = \frac{16^{\frac{3}{4}} \times (x^4)^{\frac{3}{4}}}{(y^8)^{\frac{3}{4}}}$
$= \frac{(2^4)^{\frac{3}{4}} \times x^3}{y^6} = \frac{2^3 \times x^3}{y^6} = \frac{8x^3}{y^6}$
Answer: $\frac{8x^3}{y^6}$
Example 5: Simplify $x^{\frac{2}{3}} \times x^{-\frac{1}{6}} \times x^{\frac{1}{2}}$
Add all exponents (find common denominator = 6):
$x^{\frac{4}{6} - \frac{1}{6} + \frac{3}{6}} = x^{\frac{6}{6}} = x^1 = x$
Answer: $x$
Add all exponents (find common denominator = 6):
$x^{\frac{4}{6} - \frac{1}{6} + \frac{3}{6}} = x^{\frac{6}{6}} = x^1 = x$
Answer: $x$
Example 6: Simplify $\frac{(8x^6)^{\frac{2}{3}}}{(2x)^2}$
Simplify numerator:
$(8x^6)^{\frac{2}{3}} = 8^{\frac{2}{3}} \times x^4 = (\sqrt[3]{8})^2 \times x^4 = 4x^4$
Simplify denominator:
$(2x)^2 = 4x^2$
Divide:
$\frac{4x^4}{4x^2} = x^{4-2} = x^2$
Answer: $x^2$
Simplify numerator:
$(8x^6)^{\frac{2}{3}} = 8^{\frac{2}{3}} \times x^4 = (\sqrt[3]{8})^2 \times x^4 = 4x^4$
Simplify denominator:
$(2x)^2 = 4x^2$
Divide:
$\frac{4x^4}{4x^2} = x^{4-2} = x^2$
Answer: $x^2$
Example 7: Simplify $\left(x^{-\frac{2}{3}} \times x^{\frac{5}{6}}\right)^6$
Combine inside parentheses:
$\left(x^{-\frac{4}{6} + \frac{5}{6}}\right)^6 = \left(x^{\frac{1}{6}}\right)^6 = x^{\frac{1}{6} \times 6} = x^1 = x$
Answer: $x$
Combine inside parentheses:
$\left(x^{-\frac{4}{6} + \frac{5}{6}}\right)^6 = \left(x^{\frac{1}{6}}\right)^6 = x^{\frac{1}{6} \times 6} = x^1 = x$
Answer: $x$
Example 8: Simplify $\frac{a^{\frac{3}{2}} \times b^{\frac{1}{4}}}{a^{\frac{1}{2}} \times b^{-\frac{3}{4}}}$
$\frac{a^{\frac{3}{2}}}{a^{\frac{1}{2}}} \times \frac{b^{\frac{1}{4}}}{b^{-\frac{3}{4}}} = a^{\frac{3}{2} - \frac{1}{2}} \times b^{\frac{1}{4} - (-\frac{3}{4})}$
$= a^1 \times b^{\frac{1}{4} + \frac{3}{4}} = a \times b^1 = ab$
Answer: $ab$
$\frac{a^{\frac{3}{2}}}{a^{\frac{1}{2}}} \times \frac{b^{\frac{1}{4}}}{b^{-\frac{3}{4}}} = a^{\frac{3}{2} - \frac{1}{2}} \times b^{\frac{1}{4} - (-\frac{3}{4})}$
$= a^1 \times b^{\frac{1}{4} + \frac{3}{4}} = a \times b^1 = ab$
Answer: $ab$
Summary: Laws of Rational Exponents
| Law | Formula | Example |
|---|---|---|
| Definition | $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ | $8^{\frac{2}{3}} = \sqrt[3]{8^2} = 4$ |
| Product Rule | $a^{\frac{m}{n}} \times a^{\frac{p}{q}} = a^{\frac{m}{n} + \frac{p}{q}}$ | $x^{\frac{1}{2}} \times x^{\frac{1}{3}} = x^{\frac{5}{6}}$ |
| Quotient Rule | $\frac{a^{\frac{m}{n}}}{a^{\frac{p}{q}}} = a^{\frac{m}{n} - \frac{p}{q}}$ | $\frac{y^{\frac{3}{4}}}{y^{\frac{1}{4}}} = y^{\frac{1}{2}}$ |
| Power Rule | $(a^{\frac{m}{n}})^{\frac{p}{q}} = a^{\frac{mp}{nq}}$ | $(z^{\frac{2}{3}})^{\frac{3}{2}} = z^1 = z$ |
| Power of Product | $(ab)^{\frac{m}{n}} = a^{\frac{m}{n}} b^{\frac{m}{n}}$ | $(4x)^{\frac{1}{2}} = 2x^{\frac{1}{2}}$ |
| Power of Quotient | $\left(\frac{a}{b}\right)^{\frac{m}{n}} = \frac{a^{\frac{m}{n}}}{b^{\frac{m}{n}}}$ | $\left(\frac{x}{y}\right)^{\frac{1}{2}} = \frac{x^{\frac{1}{2}}}{y^{\frac{1}{2}}}$ |
| Negative Exponent | $a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}}$ | $4^{-\frac{1}{2}} = \frac{1}{2}$ |
Rational Exponents ↔ Radical Form
| Rational Exponent | Radical Form | Meaning |
|---|---|---|
| $a^{\frac{1}{2}}$ | $\sqrt{a}$ | Square root |
| $a^{\frac{1}{3}}$ | $\sqrt[3]{a}$ | Cube root |
| $a^{\frac{1}{4}}$ | $\sqrt[4]{a}$ | Fourth root |
| $a^{\frac{1}{n}}$ | $\sqrt[n]{a}$ | nth root |
| $a^{\frac{2}{3}}$ | $\sqrt[3]{a^2}$ or $(\sqrt[3]{a})^2$ | Cube root of $a$ squared |
| $a^{\frac{3}{4}}$ | $\sqrt[4]{a^3}$ or $(\sqrt[4]{a})^3$ | Fourth root of $a$ cubed |
| $a^{\frac{m}{n}}$ | $\sqrt[n]{a^m}$ or $(\sqrt[n]{a})^m$ | nth root of $a$ to the $m$th power |
Quick Reference: Perfect Powers
Perfect Squares: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144...$
Perfect Cubes: $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000...$
Powers of 2: $2, 4, 8, 16, 32, 64, 128, 256, 512, 1024...$
Powers of 3: $3, 9, 27, 81, 243, 729...$
Powers of 5: $5, 25, 125, 625...$
Useful to memorize for quick evaluation!
Perfect Cubes: $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000...$
Powers of 2: $2, 4, 8, 16, 32, 64, 128, 256, 512, 1024...$
Powers of 3: $3, 9, 27, 81, 243, 729...$
Powers of 5: $5, 25, 125, 625...$
Useful to memorize for quick evaluation!
Common Mistakes to Avoid
❌ WRONG:
• $a^{\frac{1}{2}} + a^{\frac{1}{3}} \neq a^{\frac{5}{6}}$ (Can't add different bases/exponents like this!)
• $(a + b)^{\frac{1}{2}} \neq a^{\frac{1}{2}} + b^{\frac{1}{2}}$ (Can't distribute exponent over addition)
• $a^{\frac{2}{3}} \neq \frac{a^2}{a^3}$ (Exponent is not a regular fraction!)
• $4^{\frac{1}{2}} \times 9^{\frac{1}{2}} \neq 13^{\frac{1}{2}}$ (Can't add bases!)
✓ CORRECT:
• $a^{\frac{1}{2}} \times a^{\frac{1}{3}} = a^{\frac{5}{6}}$ (Multiplication: add exponents)
• $(ab)^{\frac{1}{2}} = a^{\frac{1}{2}} \times b^{\frac{1}{2}}$ (Can distribute over multiplication)
• $a^{\frac{2}{3}} = \sqrt[3]{a^2}$ (Numerator is power, denominator is root)
• $4^{\frac{1}{2}} \times 9^{\frac{1}{2}} = (4 \times 9)^{\frac{1}{2}} = 36^{\frac{1}{2}} = 6$
• $a^{\frac{1}{2}} + a^{\frac{1}{3}} \neq a^{\frac{5}{6}}$ (Can't add different bases/exponents like this!)
• $(a + b)^{\frac{1}{2}} \neq a^{\frac{1}{2}} + b^{\frac{1}{2}}$ (Can't distribute exponent over addition)
• $a^{\frac{2}{3}} \neq \frac{a^2}{a^3}$ (Exponent is not a regular fraction!)
• $4^{\frac{1}{2}} \times 9^{\frac{1}{2}} \neq 13^{\frac{1}{2}}$ (Can't add bases!)
✓ CORRECT:
• $a^{\frac{1}{2}} \times a^{\frac{1}{3}} = a^{\frac{5}{6}}$ (Multiplication: add exponents)
• $(ab)^{\frac{1}{2}} = a^{\frac{1}{2}} \times b^{\frac{1}{2}}$ (Can distribute over multiplication)
• $a^{\frac{2}{3}} = \sqrt[3]{a^2}$ (Numerator is power, denominator is root)
• $4^{\frac{1}{2}} \times 9^{\frac{1}{2}} = (4 \times 9)^{\frac{1}{2}} = 36^{\frac{1}{2}} = 6$
Success Tips for Rational Exponents:
✓ Remember: numerator = power, denominator = root
✓ Take the root first (usually easier with smaller numbers)
✓ Memorize perfect squares, cubes, and powers
✓ All integer exponent rules still apply!
✓ Negative rational exponent = reciprocal
✓ Find common denominators when adding/subtracting exponents
✓ Always simplify fractions in exponents
✓ Convert to radical form if it helps visualize
✓ Check your work by converting back to radical form
✓ Practice recognizing equivalent forms
✓ Remember: numerator = power, denominator = root
✓ Take the root first (usually easier with smaller numbers)
✓ Memorize perfect squares, cubes, and powers
✓ All integer exponent rules still apply!
✓ Negative rational exponent = reciprocal
✓ Find common denominators when adding/subtracting exponents
✓ Always simplify fractions in exponents
✓ Convert to radical form if it helps visualize
✓ Check your work by converting back to radical form
✓ Practice recognizing equivalent forms