Rational Exponents

Converting between radicals and rational exponents

Rational Exponents

Definition

A rational exponent is a fraction exponent:

$a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

Special Cases

$a^{\frac{1}{n}} = \sqrt[n]{a}$

Examples:

$x^{\frac{1}{2}} = \sqrt{x}$
$x^{\frac{1}{3}} = \sqrt[3]{x}$
$8^{\frac{1}{3}} = \sqrt[3]{8} = 2$

General Form

$a^{\frac{m}{n}}$

Numerator ( $m$ ): power
Denominator ( $n$ ): root

Example: $27^{\frac{2}{3}}$

Method 1: $27^{\frac{2}{3}} = (\sqrt[3]{27})^2 = 3^2 = 9$

Method 2: $27^{\frac{2}{3}} = \sqrt[3]{27^2} = \sqrt[3]{729} = 9$

Negative Rational Exponents

$a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}}$

Example: $16^{-\frac{3}{4}} = \frac{1}{16^{\frac{3}{4}}} = \frac{1}{(\sqrt[4]{16})^3} = \frac{1}{2^3} = \frac{1}{8}$$

All Exponent Rules Apply!

Product: $a^m \cdot a^n = a^{m+n}$
Quotient: $\frac{a^m}{a^n} = a^{m-n}$
Power: $(a^m)^n = a^{mn}$

📚 Practice Problems

1Problem 1easy

❓ Question:

Evaluate: 16^(1/2)

💡 Show Solution

Step 1: Understand the notation: a^(1/n) = ⁿ√a

Step 2: Apply to our problem: 16^(1/2) = √16

Step 3: Calculate: √16 = 4

Answer: 4

2Problem 2easy

❓ Question:

Evaluate: $25^{\frac{1}{2}}$

💡 Show Solution

$25^{\frac{1}{2}} = \sqrt{25} = 5$

Answer: $5$

3Problem 3easy

❓ Question:

Simplify: 8^(2/3)

💡 Show Solution

Step 1: Understand the notation: a^(m/n) = (ⁿ√a)^m or ⁿ√(a^m)

Step 2: Choose the easier approach: 8^(2/3) = (∛8)²

Step 3: Find the cube root: ∛8 = 2

Step 4: Square the result: 2² = 4

Answer: 4

4Problem 4medium

❓ Question:

Simplify: $16^{\frac{3}{4}}$

💡 Show Solution

Method 1: Take the root first, then the power $16^{\frac{3}{4}} = (\sqrt[4]{16})^3 = 2^3 = 8$

Method 2: Power first, then root $16^{\frac{3}{4}} = \sqrt[4]{16^3} = \sqrt[4]{4096} = 8$

Answer: $8$

5Problem 5medium

❓ Question:

Simplify: (x^(3/4))(x^(1/4))

💡 Show Solution

Step 1: Use product rule: x^m · x^n = x^(m+n)

Step 2: Add the exponents: x^(3/4) · x^(1/4) = x^(3/4 + 1/4)

Step 3: Add fractions: 3/4 + 1/4 = 4/4 = 1

Step 4: Simplify: x^1 = x

Answer: x

6Problem 6medium

❓ Question:

Solve: x^(3/2) = 27

💡 Show Solution

Step 1: Raise both sides to the reciprocal power: The reciprocal of 3/2 is 2/3 (x^(3/2))^(2/3) = 27^(2/3)

Step 2: Simplify left side using power rule: x^((3/2)·(2/3)) = 27^(2/3) x^1 = 27^(2/3) x = 27^(2/3)

Step 3: Evaluate 27^(2/3): 27^(2/3) = (∛27)² = 3² = 9

Step 4: Check: 9^(3/2) = (√9)³ = 3³ = 27 ✓

Answer: x = 9

7Problem 7hard

❓ Question:

Simplify: $\frac{x^{\frac{5}{3}}}{x^{\frac{2}{3}}}$

💡 Show Solution

Use the quotient rule: $\frac{a^m}{a^n} = a^{m-n}$

$\frac{x^{\frac{5}{3}}}{x^{\frac{2}{3}}} = x^{\frac{5}{3} - \frac{2}{3}}$

$= x^{\frac{3}{3}}$

$= x^1 = x$

Answer: $x$

8Problem 8hard

❓ Question:

Simplify: (16x^8y^4)^(3/4)

💡 Show Solution

Step 1: Apply power to each factor: (16x^8y^4)^(3/4) = 16^(3/4) · (x^8)^(3/4) · (y^4)^(3/4)

Step 2: Evaluate 16^(3/4): 16^(3/4) = (⁴√16)³ = 2³ = 8

Step 3: Apply power rule to x: (x^8)^(3/4) = x^(8·3/4) = x^(24/4) = x^6

Step 4: Apply power rule to y: (y^4)^(3/4) = y^(4·3/4) = y^(12/4) = y^3

Step 5: Combine: 8x^6y^3

Answer: 8x^6y^3

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