Integer Exponents - Ninth Grade Math
Introduction to Exponents
Exponent (Power): A number that indicates how many times the base is multiplied by itself
Base: The number being multiplied
Exponential Form: $a^n$ where $a$ is the base and $n$ is the exponent
Read as: "$a$ to the power of $n$" or "$a$ raised to the $n$th power"
Base: The number being multiplied
Exponential Form: $a^n$ where $a$ is the base and $n$ is the exponent
Read as: "$a$ to the power of $n$" or "$a$ raised to the $n$th power"
Exponential Notation:
$$a^n = \underbrace{a \times a \times a \times \cdots \times a}_{n \text{ times}}$$
Examples:
• $2^3 = 2 \times 2 \times 2 = 8$
• $5^4 = 5 \times 5 \times 5 \times 5 = 625$
• $10^2 = 10 \times 10 = 100$
$$a^n = \underbrace{a \times a \times a \times \cdots \times a}_{n \text{ times}}$$
Examples:
• $2^3 = 2 \times 2 \times 2 = 8$
• $5^4 = 5 \times 5 \times 5 \times 5 = 625$
• $10^2 = 10 \times 10 = 100$
1-2. Powers with Integer, Decimal, and Fractional Bases
Powers with Integer Bases
Positive Integer Bases:
• $3^4 = 3 \times 3 \times 3 \times 3 = 81$
• $7^2 = 7 \times 7 = 49$
Negative Integer Bases:
• Even exponent: Result is POSITIVE
$(-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = 16$
• Odd exponent: Result is NEGATIVE
$(-2)^3 = (-2) \times (-2) \times (-2) = -8$
• $3^4 = 3 \times 3 \times 3 \times 3 = 81$
• $7^2 = 7 \times 7 = 49$
Negative Integer Bases:
• Even exponent: Result is POSITIVE
$(-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = 16$
• Odd exponent: Result is NEGATIVE
$(-2)^3 = (-2) \times (-2) \times (-2) = -8$
Important: Parentheses Matter!
• $(-3)^2 = (-3) \times (-3) = 9$ ✓
• $-3^2 = -(3 \times 3) = -9$ (negative is not squared!)
• $(-3)^2 = (-3) \times (-3) = 9$ ✓
• $-3^2 = -(3 \times 3) = -9$ (negative is not squared!)
Example 1: Evaluate $(-5)^3$
$(-5)^3 = (-5) \times (-5) \times (-5) = 25 \times (-5) = -125$
Answer: $-125$
$(-5)^3 = (-5) \times (-5) \times (-5) = 25 \times (-5) = -125$
Answer: $-125$
Example 2: Evaluate $(-4)^4$
$(-4)^4 = (-4) \times (-4) \times (-4) \times (-4) = 16 \times 16 = 256$
Answer: $256$
$(-4)^4 = (-4) \times (-4) \times (-4) \times (-4) = 16 \times 16 = 256$
Answer: $256$
Powers with Decimal Bases
Example 3: Evaluate $(0.5)^3$
$(0.5)^3 = 0.5 \times 0.5 \times 0.5 = 0.25 \times 0.5 = 0.125$
Answer: $0.125$
$(0.5)^3 = 0.5 \times 0.5 \times 0.5 = 0.25 \times 0.5 = 0.125$
Answer: $0.125$
Example 4: Evaluate $(1.2)^2$
$(1.2)^2 = 1.2 \times 1.2 = 1.44$
Answer: $1.44$
$(1.2)^2 = 1.2 \times 1.2 = 1.44$
Answer: $1.44$
Powers with Fractional Bases
Rule for Fractions:
$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$
Raise both numerator and denominator to the power
$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$
Raise both numerator and denominator to the power
Example 5: Evaluate $\left(\frac{2}{3}\right)^3$
$\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}$
Answer: $\frac{8}{27}$
$\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}$
Answer: $\frac{8}{27}$
Example 6: Evaluate $\left(\frac{-3}{4}\right)^2$
$\left(\frac{-3}{4}\right)^2 = \frac{(-3)^2}{4^2} = \frac{9}{16}$
Answer: $\frac{9}{16}$
$\left(\frac{-3}{4}\right)^2 = \frac{(-3)^2}{4^2} = \frac{9}{16}$
Answer: $\frac{9}{16}$
3. Negative Exponents
Negative Exponent: Represents the reciprocal of the base raised to the positive exponent
Negative Exponent Rule:
$$a^{-n} = \frac{1}{a^n}$$
where $a \neq 0$
Conversely:
$$\frac{1}{a^{-n}} = a^n$$
$$a^{-n} = \frac{1}{a^n}$$
where $a \neq 0$
Conversely:
$$\frac{1}{a^{-n}} = a^n$$
Key Understanding:
• Negative exponent means "take the reciprocal"
• Move the base from numerator to denominator (or vice versa)
• Change the sign of the exponent when moving
• The base itself doesn't become negative!
• Negative exponent means "take the reciprocal"
• Move the base from numerator to denominator (or vice versa)
• Change the sign of the exponent when moving
• The base itself doesn't become negative!
Example 1: Simplify $2^{-3}$
$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$
Answer: $\frac{1}{8}$
$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$
Answer: $\frac{1}{8}$
Example 2: Simplify $5^{-2}$
$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$
Answer: $\frac{1}{25}$
$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$
Answer: $\frac{1}{25}$
Example 3: Simplify $\frac{1}{3^{-4}}$
$\frac{1}{3^{-4}} = 3^4 = 81$
Answer: $81$
$\frac{1}{3^{-4}} = 3^4 = 81$
Answer: $81$
Example 4: Simplify $(-2)^{-3}$
$(-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8} = -\frac{1}{8}$
Answer: $-\frac{1}{8}$
$(-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8} = -\frac{1}{8}$
Answer: $-\frac{1}{8}$
Example 5: Simplify $\left(\frac{2}{5}\right)^{-2}$
$\left(\frac{2}{5}\right)^{-2} = \left(\frac{5}{2}\right)^{2} = \frac{25}{4}$
Answer: $\frac{25}{4}$
$\left(\frac{2}{5}\right)^{-2} = \left(\frac{5}{2}\right)^{2} = \frac{25}{4}$
Answer: $\frac{25}{4}$
4-6. Laws of Exponents
Seven Fundamental Laws of Exponents:
1. Product Rule (Multiplication):
$$a^m \times a^n = a^{m+n}$$
2. Quotient Rule (Division):
$$\frac{a^m}{a^n} = a^{m-n}$$
3. Power Rule (Power of a Power):
$$(a^m)^n = a^{m \times n}$$
4. Power of a Product:
$$(ab)^n = a^n \times b^n$$
5. Power of a Quotient:
$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$
6. Zero Exponent Rule:
$$a^0 = 1$$ (where $a \neq 0$)
7. Negative Exponent Rule:
$$a^{-n} = \frac{1}{a^n}$$
1. Product Rule (Multiplication):
$$a^m \times a^n = a^{m+n}$$
2. Quotient Rule (Division):
$$\frac{a^m}{a^n} = a^{m-n}$$
3. Power Rule (Power of a Power):
$$(a^m)^n = a^{m \times n}$$
4. Power of a Product:
$$(ab)^n = a^n \times b^n$$
5. Power of a Quotient:
$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$
6. Zero Exponent Rule:
$$a^0 = 1$$ (where $a \neq 0$)
7. Negative Exponent Rule:
$$a^{-n} = \frac{1}{a^n}$$
4. Multiplication Rule (Product Rule)
Product Rule: When multiplying powers with the same base, add the exponents
Product Rule Formula:
$$a^m \times a^n = a^{m+n}$$
Why it works:
$a^3 \times a^2 = (a \times a \times a) \times (a \times a) = a^5$
$$a^m \times a^n = a^{m+n}$$
Why it works:
$a^3 \times a^2 = (a \times a \times a) \times (a \times a) = a^5$
Example 1: Simplify $3^4 \times 3^2$
$3^4 \times 3^2 = 3^{4+2} = 3^6 = 729$
Answer: $3^6$ or $729$
$3^4 \times 3^2 = 3^{4+2} = 3^6 = 729$
Answer: $3^6$ or $729$
Example 2: Simplify $x^5 \times x^3$
$x^5 \times x^3 = x^{5+3} = x^8$
Answer: $x^8$
$x^5 \times x^3 = x^{5+3} = x^8$
Answer: $x^8$
Example 3: Simplify $2^7 \times 2^{-3}$
$2^7 \times 2^{-3} = 2^{7+(-3)} = 2^{7-3} = 2^4 = 16$
Answer: $2^4$ or $16$
$2^7 \times 2^{-3} = 2^{7+(-3)} = 2^{7-3} = 2^4 = 16$
Answer: $2^4$ or $16$
Example 4: Simplify $a^{-2} \times a^5$
$a^{-2} \times a^5 = a^{-2+5} = a^3$
Answer: $a^3$
$a^{-2} \times a^5 = a^{-2+5} = a^3$
Answer: $a^3$
Important: Product rule ONLY works with the SAME base!
• $3^2 \times 3^4 = 3^6$ ✓
• $2^3 \times 3^2$ CANNOT be simplified using product rule ✗
• $3^2 \times 3^4 = 3^6$ ✓
• $2^3 \times 3^2$ CANNOT be simplified using product rule ✗
5. Division Rule (Quotient Rule)
Quotient Rule: When dividing powers with the same base, subtract the exponents
Quotient Rule Formula:
$$\frac{a^m}{a^n} = a^{m-n}$$
Why it works:
$\frac{a^5}{a^2} = \frac{a \times a \times a \times a \times a}{a \times a} = a \times a \times a = a^3 = a^{5-2}$
$$\frac{a^m}{a^n} = a^{m-n}$$
Why it works:
$\frac{a^5}{a^2} = \frac{a \times a \times a \times a \times a}{a \times a} = a \times a \times a = a^3 = a^{5-2}$
Example 1: Simplify $\frac{5^7}{5^3}$
$\frac{5^7}{5^3} = 5^{7-3} = 5^4 = 625$
Answer: $5^4$ or $625$
$\frac{5^7}{5^3} = 5^{7-3} = 5^4 = 625$
Answer: $5^4$ or $625$
Example 2: Simplify $\frac{x^8}{x^5}$
$\frac{x^8}{x^5} = x^{8-5} = x^3$
Answer: $x^3$
$\frac{x^8}{x^5} = x^{8-5} = x^3$
Answer: $x^3$
Example 3: Simplify $\frac{3^4}{3^6}$
$\frac{3^4}{3^6} = 3^{4-6} = 3^{-2} = \frac{1}{3^2} = \frac{1}{9}$
Answer: $3^{-2}$ or $\frac{1}{9}$
$\frac{3^4}{3^6} = 3^{4-6} = 3^{-2} = \frac{1}{3^2} = \frac{1}{9}$
Answer: $3^{-2}$ or $\frac{1}{9}$
Example 4: Simplify $\frac{y^3}{y^{-2}}$
$\frac{y^3}{y^{-2}} = y^{3-(-2)} = y^{3+2} = y^5$
Answer: $y^5$
$\frac{y^3}{y^{-2}} = y^{3-(-2)} = y^{3+2} = y^5$
Answer: $y^5$
Example 5: Simplify $\frac{7^5}{7^5}$
$\frac{7^5}{7^5} = 7^{5-5} = 7^0 = 1$
Answer: $1$
$\frac{7^5}{7^5} = 7^{5-5} = 7^0 = 1$
Answer: $1$
6. Power Rule (Power of a Power)
Power Rule: When raising a power to another power, multiply the exponents
Power Rule Formula:
$$(a^m)^n = a^{m \times n}$$
Why it works:
$(a^2)^3 = a^2 \times a^2 \times a^2 = a^{2+2+2} = a^6 = a^{2 \times 3}$
$$(a^m)^n = a^{m \times n}$$
Why it works:
$(a^2)^3 = a^2 \times a^2 \times a^2 = a^{2+2+2} = a^6 = a^{2 \times 3}$
Example 1: Simplify $(2^3)^4$
$(2^3)^4 = 2^{3 \times 4} = 2^{12} = 4096$
Answer: $2^{12}$ or $4096$
$(2^3)^4 = 2^{3 \times 4} = 2^{12} = 4096$
Answer: $2^{12}$ or $4096$
Example 2: Simplify $(x^5)^3$
$(x^5)^3 = x^{5 \times 3} = x^{15}$
Answer: $x^{15}$
$(x^5)^3 = x^{5 \times 3} = x^{15}$
Answer: $x^{15}$
Example 3: Simplify $(3^{-2})^4$
$(3^{-2})^4 = 3^{-2 \times 4} = 3^{-8} = \frac{1}{3^8}$
Answer: $3^{-8}$ or $\frac{1}{6561}$
$(3^{-2})^4 = 3^{-2 \times 4} = 3^{-8} = \frac{1}{3^8}$
Answer: $3^{-8}$ or $\frac{1}{6561}$
Example 4: Simplify $(y^4)^{-2}$
$(y^4)^{-2} = y^{4 \times (-2)} = y^{-8} = \frac{1}{y^8}$
Answer: $y^{-8}$ or $\frac{1}{y^8}$
$(y^4)^{-2} = y^{4 \times (-2)} = y^{-8} = \frac{1}{y^8}$
Answer: $y^{-8}$ or $\frac{1}{y^8}$
Power of a Product and Quotient
Power of a Product:
$$(ab)^n = a^n \times b^n$$
Power of a Quotient:
$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$
$$(ab)^n = a^n \times b^n$$
Power of a Quotient:
$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$
Example: Simplify $(2x)^3$
$(2x)^3 = 2^3 \times x^3 = 8x^3$
Answer: $8x^3$
$(2x)^3 = 2^3 \times x^3 = 8x^3$
Answer: $8x^3$
Example: Simplify $\left(\frac{3x}{y}\right)^2$
$\left(\frac{3x}{y}\right)^2 = \frac{(3x)^2}{y^2} = \frac{9x^2}{y^2}$
Answer: $\frac{9x^2}{y^2}$
$\left(\frac{3x}{y}\right)^2 = \frac{(3x)^2}{y^2} = \frac{9x^2}{y^2}$
Answer: $\frac{9x^2}{y^2}$
7-11. Simplify Exponential Expressions
Simplifying: Using exponent rules to write expressions in simplest form
Simplified Form: Positive exponents, no parentheses, reduced expressions
Simplified Form: Positive exponents, no parentheses, reduced expressions
7. Using Multiplication Rule
Example 1: Simplify $3^4 \times 3^6 \times 3^2$
$3^4 \times 3^6 \times 3^2 = 3^{4+6+2} = 3^{12}$
Answer: $3^{12}$
$3^4 \times 3^6 \times 3^2 = 3^{4+6+2} = 3^{12}$
Answer: $3^{12}$
Example 2: Simplify $x^3 \times x \times x^5$
$x^3 \times x^1 \times x^5 = x^{3+1+5} = x^9$
Answer: $x^9$
$x^3 \times x^1 \times x^5 = x^{3+1+5} = x^9$
Answer: $x^9$
Example 3: Simplify $2^5 \times 3^2 \times 2^3$
$2^5 \times 2^3 \times 3^2 = 2^{5+3} \times 3^2 = 2^8 \times 3^2$
Answer: $2^8 \times 3^2$ or $256 \times 9 = 2304$
$2^5 \times 2^3 \times 3^2 = 2^{5+3} \times 3^2 = 2^8 \times 3^2$
Answer: $2^8 \times 3^2$ or $256 \times 9 = 2304$
8. Using Division Rule
Example 1: Simplify $\frac{x^{10}}{x^7}$
$\frac{x^{10}}{x^7} = x^{10-7} = x^3$
Answer: $x^3$
$\frac{x^{10}}{x^7} = x^{10-7} = x^3$
Answer: $x^3$
Example 2: Simplify $\frac{5^8}{5^3}$
$\frac{5^8}{5^3} = 5^{8-3} = 5^5 = 3125$
Answer: $5^5$ or $3125$
$\frac{5^8}{5^3} = 5^{8-3} = 5^5 = 3125$
Answer: $5^5$ or $3125$
Example 3: Simplify $\frac{a^4 \times a^6}{a^7}$
$\frac{a^{4+6}}{a^7} = \frac{a^{10}}{a^7} = a^{10-7} = a^3$
Answer: $a^3$
$\frac{a^{4+6}}{a^7} = \frac{a^{10}}{a^7} = a^{10-7} = a^3$
Answer: $a^3$
9. Using Multiplication and Division Rules
Example 1: Simplify $\frac{x^5 \times x^3}{x^2}$
$\frac{x^5 \times x^3}{x^2} = \frac{x^{5+3}}{x^2} = \frac{x^8}{x^2} = x^{8-2} = x^6$
Answer: $x^6$
$\frac{x^5 \times x^3}{x^2} = \frac{x^{5+3}}{x^2} = \frac{x^8}{x^2} = x^{8-2} = x^6$
Answer: $x^6$
Example 2: Simplify $\frac{2^7 \times 2^{-3}}{2^2}$
$\frac{2^7 \times 2^{-3}}{2^2} = \frac{2^{7-3}}{2^2} = \frac{2^4}{2^2} = 2^{4-2} = 2^2 = 4$
Answer: $2^2$ or $4$
$\frac{2^7 \times 2^{-3}}{2^2} = \frac{2^{7-3}}{2^2} = \frac{2^4}{2^2} = 2^{4-2} = 2^2 = 4$
Answer: $2^2$ or $4$
Example 3: Simplify $\frac{3^6 \times 5^4}{3^2 \times 5^2}$
$\frac{3^6 \times 5^4}{3^2 \times 5^2} = 3^{6-2} \times 5^{4-2} = 3^4 \times 5^2 = 81 \times 25 = 2025$
Answer: $3^4 \times 5^2$ or $2025$
$\frac{3^6 \times 5^4}{3^2 \times 5^2} = 3^{6-2} \times 5^{4-2} = 3^4 \times 5^2 = 81 \times 25 = 2025$
Answer: $3^4 \times 5^2$ or $2025$
10. Using Power Rule
Example 1: Simplify $(x^3)^5$
$(x^3)^5 = x^{3 \times 5} = x^{15}$
Answer: $x^{15}$
$(x^3)^5 = x^{3 \times 5} = x^{15}$
Answer: $x^{15}$
Example 2: Simplify $(2^2)^3 \times 2^4$
$(2^2)^3 \times 2^4 = 2^{2 \times 3} \times 2^4 = 2^6 \times 2^4 = 2^{6+4} = 2^{10} = 1024$
Answer: $2^{10}$ or $1024$
$(2^2)^3 \times 2^4 = 2^{2 \times 3} \times 2^4 = 2^6 \times 2^4 = 2^{6+4} = 2^{10} = 1024$
Answer: $2^{10}$ or $1024$
Example 3: Simplify $\left(\frac{x^4}{x}\right)^3$
$\left(\frac{x^4}{x}\right)^3 = (x^{4-1})^3 = (x^3)^3 = x^{3 \times 3} = x^9$
Answer: $x^9$
$\left(\frac{x^4}{x}\right)^3 = (x^{4-1})^3 = (x^3)^3 = x^{3 \times 3} = x^9$
Answer: $x^9$
11. Using All Exponent Rules
Example 1: Simplify $\frac{(x^2)^3 \times x^4}{x^5}$
$\frac{(x^2)^3 \times x^4}{x^5} = \frac{x^6 \times x^4}{x^5} = \frac{x^{10}}{x^5} = x^5$
Answer: $x^5$
$\frac{(x^2)^3 \times x^4}{x^5} = \frac{x^6 \times x^4}{x^5} = \frac{x^{10}}{x^5} = x^5$
Answer: $x^5$
Example 2: Simplify $(3x^2y)^3$
$(3x^2y)^3 = 3^3 \times (x^2)^3 \times y^3 = 27x^6y^3$
Answer: $27x^6y^3$
$(3x^2y)^3 = 3^3 \times (x^2)^3 \times y^3 = 27x^6y^3$
Answer: $27x^6y^3$
Example 3: Simplify $\frac{(2a^3)^2 \times a^4}{4a^5}$
$\frac{(2a^3)^2 \times a^4}{4a^5} = \frac{4a^6 \times a^4}{4a^5} = \frac{4a^{10}}{4a^5} = a^5$
Answer: $a^5$
$\frac{(2a^3)^2 \times a^4}{4a^5} = \frac{4a^6 \times a^4}{4a^5} = \frac{4a^{10}}{4a^5} = a^5$
Answer: $a^5$
Example 4: Simplify $\left(\frac{x^{-2} \times x^5}{x^2}\right)^{-1}$
$\left(\frac{x^{-2+5}}{x^2}\right)^{-1} = \left(\frac{x^3}{x^2}\right)^{-1} = (x^1)^{-1} = x^{-1} = \frac{1}{x}$
Answer: $\frac{1}{x}$
$\left(\frac{x^{-2+5}}{x^2}\right)^{-1} = \left(\frac{x^3}{x^2}\right)^{-1} = (x^1)^{-1} = x^{-1} = \frac{1}{x}$
Answer: $\frac{1}{x}$
12. Evaluate Expressions Using Exponent Rules
Evaluating: Finding the numerical value of an expression using exponent rules
Example 1: Evaluate $\frac{3^5 \times 3^2}{3^4}$
$\frac{3^5 \times 3^2}{3^4} = \frac{3^{5+2}}{3^4} = \frac{3^7}{3^4} = 3^{7-4} = 3^3 = 27$
Answer: $27$
$\frac{3^5 \times 3^2}{3^4} = \frac{3^{5+2}}{3^4} = \frac{3^7}{3^4} = 3^{7-4} = 3^3 = 27$
Answer: $27$
Example 2: Evaluate $(2^3)^2 \times 2^{-4}$
$(2^3)^2 \times 2^{-4} = 2^6 \times 2^{-4} = 2^{6-4} = 2^2 = 4$
Answer: $4$
$(2^3)^2 \times 2^{-4} = 2^6 \times 2^{-4} = 2^{6-4} = 2^2 = 4$
Answer: $4$
Example 3: Evaluate $\frac{5^4 \times 5^{-2}}{5^0}$
$\frac{5^4 \times 5^{-2}}{5^0} = \frac{5^{4-2}}{1} = 5^2 = 25$
Answer: $25$
$\frac{5^4 \times 5^{-2}}{5^0} = \frac{5^{4-2}}{1} = 5^2 = 25$
Answer: $25$
Example 4: Evaluate $\left(\frac{2^3}{2^5}\right)^2$
$\left(\frac{2^3}{2^5}\right)^2 = (2^{3-5})^2 = (2^{-2})^2 = 2^{-4} = \frac{1}{16}$
Answer: $\frac{1}{16}$
$\left(\frac{2^3}{2^5}\right)^2 = (2^{3-5})^2 = (2^{-2})^2 = 2^{-4} = \frac{1}{16}$
Answer: $\frac{1}{16}$
13-14. Identify Equivalent Exponential Expressions
Equivalent Expressions: Different expressions that have the same value or simplify to the same form
How to Identify Equivalent Expressions:
1. Simplify each expression using exponent rules
2. Compare simplified forms
3. If simplified forms match → expressions are equivalent
4. Can also substitute values and compare results
1. Simplify each expression using exponent rules
2. Compare simplified forms
3. If simplified forms match → expressions are equivalent
4. Can also substitute values and compare results
Example 1: Are $x^3 \times x^4$ and $x^7$ equivalent?
Simplify: $x^3 \times x^4 = x^{3+4} = x^7$ ✓
Answer: YES, they are equivalent
Simplify: $x^3 \times x^4 = x^{3+4} = x^7$ ✓
Answer: YES, they are equivalent
Example 2: Are $(2^3)^2$ and $2^5$ equivalent?
Simplify: $(2^3)^2 = 2^{3 \times 2} = 2^6$
Compare: $2^6 \neq 2^5$
Answer: NO, they are not equivalent
Simplify: $(2^3)^2 = 2^{3 \times 2} = 2^6$
Compare: $2^6 \neq 2^5$
Answer: NO, they are not equivalent
Example 3: Which expressions are equivalent to $\frac{x^8}{x^2}$?
a) $x^6$ b) $x^4$ c) $(x^3)^2$ d) $x^3 \times x^3$
Simplify original: $\frac{x^8}{x^2} = x^{8-2} = x^6$
Check options:
a) $x^6$ ✓ Matches!
b) $x^4$ ✗ Different
c) $(x^3)^2 = x^6$ ✓ Matches!
d) $x^3 \times x^3 = x^6$ ✓ Matches!
Answer: Options a, c, and d are equivalent
a) $x^6$ b) $x^4$ c) $(x^3)^2$ d) $x^3 \times x^3$
Simplify original: $\frac{x^8}{x^2} = x^{8-2} = x^6$
Check options:
a) $x^6$ ✓ Matches!
b) $x^4$ ✗ Different
c) $(x^3)^2 = x^6$ ✓ Matches!
d) $x^3 \times x^3 = x^6$ ✓ Matches!
Answer: Options a, c, and d are equivalent
Example 4: Match equivalent expressions:
Column A:
1. $3^5 \times 3^2$
2. $(3^2)^3$
3. $\frac{3^{10}}{3^3}$
4. $3^4 \times 3^3$
Column B:
A. $3^6$
B. $3^7$
C. $3^8$
Solutions:
1. $3^5 \times 3^2 = 3^7$ → matches B
2. $(3^2)^3 = 3^6$ → matches A
3. $\frac{3^{10}}{3^3} = 3^7$ → matches B
4. $3^4 \times 3^3 = 3^7$ → matches B
Column A:
1. $3^5 \times 3^2$
2. $(3^2)^3$
3. $\frac{3^{10}}{3^3}$
4. $3^4 \times 3^3$
Column B:
A. $3^6$
B. $3^7$
C. $3^8$
Solutions:
1. $3^5 \times 3^2 = 3^7$ → matches B
2. $(3^2)^3 = 3^6$ → matches A
3. $\frac{3^{10}}{3^3} = 3^7$ → matches B
4. $3^4 \times 3^3 = 3^7$ → matches B
Quick Reference: Laws of Exponents
| Law | Formula | Example |
|---|---|---|
| Product Rule | $a^m \times a^n = a^{m+n}$ | $x^3 \times x^5 = x^8$ |
| Quotient Rule | $\frac{a^m}{a^n} = a^{m-n}$ | $\frac{y^7}{y^3} = y^4$ |
| Power Rule | $(a^m)^n = a^{m \times n}$ | $(z^2)^4 = z^8$ |
| Power of Product | $(ab)^n = a^n b^n$ | $(2x)^3 = 8x^3$ |
| Power of Quotient | $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ | $\left(\frac{x}{y}\right)^2 = \frac{x^2}{y^2}$ |
| Zero Exponent | $a^0 = 1$ | $5^0 = 1$ |
| Negative Exponent | $a^{-n} = \frac{1}{a^n}$ | $2^{-3} = \frac{1}{8}$ |
| Identity | $a^1 = a$ | $7^1 = 7$ |
Special Cases and Important Notes
Zero Exponent ($a^0 = 1$):
Any non-zero base raised to the power of 0 equals 1
• $7^0 = 1$
• $(-5)^0 = 1$
• $(x^3)^0 = 1$
• Note: $0^0$ is undefined
Any non-zero base raised to the power of 0 equals 1
• $7^0 = 1$
• $(-5)^0 = 1$
• $(x^3)^0 = 1$
• Note: $0^0$ is undefined
One Exponent ($a^1 = a$):
Any base raised to the power of 1 equals itself
• $9^1 = 9$
• $x^1 = x$
Any base raised to the power of 1 equals itself
• $9^1 = 9$
• $x^1 = x$
Negative Base Rules:
• Even exponent: Result is POSITIVE
$(-3)^4 = 81$
• Odd exponent: Result is NEGATIVE
$(-3)^3 = -27$
• Watch parentheses!
$(-2)^2 = 4$ but $-2^2 = -4$
• Even exponent: Result is POSITIVE
$(-3)^4 = 81$
• Odd exponent: Result is NEGATIVE
$(-3)^3 = -27$
• Watch parentheses!
$(-2)^2 = 4$ but $-2^2 = -4$
Common Mistakes to Avoid:
• $x^2 + x^3 \neq x^5$ (Different operations - can't combine!)
• $(x^2)^3 \neq x^5$ (Should be $x^6$)
• $2x^3 \neq (2x)^3$ ($2x^3 = 2 \times x^3$, but $(2x)^3 = 8x^3$)
• $x^{-2} \neq -x^2$ ($x^{-2} = \frac{1}{x^2}$)
• $\frac{1}{2^{-3}} \neq \frac{1}{8}$ (Should be $8$)
• $x^2 + x^3 \neq x^5$ (Different operations - can't combine!)
• $(x^2)^3 \neq x^5$ (Should be $x^6$)
• $2x^3 \neq (2x)^3$ ($2x^3 = 2 \times x^3$, but $(2x)^3 = 8x^3$)
• $x^{-2} \neq -x^2$ ($x^{-2} = \frac{1}{x^2}$)
• $\frac{1}{2^{-3}} \neq \frac{1}{8}$ (Should be $8$)
Step-by-Step Simplification Strategy
Order of Operations for Simplifying:
Step 1: Deal with parentheses first (power rule)
Step 2: Apply negative exponent rule (move to reciprocal)
Step 3: Apply product rule (add exponents when multiplying)
Step 4: Apply quotient rule (subtract exponents when dividing)
Step 5: Simplify coefficients
Step 6: Write with positive exponents
Step 7: Verify no further simplification possible
Step 1: Deal with parentheses first (power rule)
Step 2: Apply negative exponent rule (move to reciprocal)
Step 3: Apply product rule (add exponents when multiplying)
Step 4: Apply quotient rule (subtract exponents when dividing)
Step 5: Simplify coefficients
Step 6: Write with positive exponents
Step 7: Verify no further simplification possible
Complex Example: Simplify $\frac{(2x^{-3})^2 \times x^7}{4x^{-2}}$
Step 1: Power rule
$\frac{2^2 \times (x^{-3})^2 \times x^7}{4x^{-2}} = \frac{4x^{-6} \times x^7}{4x^{-2}}$
Step 2: Product rule in numerator
$\frac{4x^{-6+7}}{4x^{-2}} = \frac{4x^1}{4x^{-2}}$
Step 3: Simplify coefficient and quotient rule
$\frac{4x}{4x^{-2}} = x^{1-(-2)} = x^{1+2} = x^3$
Answer: $x^3$
Step 1: Power rule
$\frac{2^2 \times (x^{-3})^2 \times x^7}{4x^{-2}} = \frac{4x^{-6} \times x^7}{4x^{-2}}$
Step 2: Product rule in numerator
$\frac{4x^{-6+7}}{4x^{-2}} = \frac{4x^1}{4x^{-2}}$
Step 3: Simplify coefficient and quotient rule
$\frac{4x}{4x^{-2}} = x^{1-(-2)} = x^{1+2} = x^3$
Answer: $x^3$
Success Tips for Integer Exponents:
✓ Master the seven basic laws of exponents
✓ Remember: same base is required for product and quotient rules
✓ Negative exponent means reciprocal, NOT negative number
✓ When in doubt, expand and count the factors
✓ Always write final answers with positive exponents
✓ Watch parentheses carefully with negative bases
✓ Practice identifying equivalent expressions
✓ Check your work by substituting simple values
✓ Remember: $a^0 = 1$ for any $a \neq 0$
✓ Keep track of signs when dealing with negative bases
✓ Master the seven basic laws of exponents
✓ Remember: same base is required for product and quotient rules
✓ Negative exponent means reciprocal, NOT negative number
✓ When in doubt, expand and count the factors
✓ Always write final answers with positive exponents
✓ Watch parentheses carefully with negative bases
✓ Practice identifying equivalent expressions
✓ Check your work by substituting simple values
✓ Remember: $a^0 = 1$ for any $a \neq 0$
✓ Keep track of signs when dealing with negative bases