Understanding how to rewrite expressions with roots and powers is essential for simplifying complex equations. Below is a breakdown of the core concepts using the example: $\frac{1}{(\sqrt{2})^5}$.
1. The Core Identities
To master these expressions, you need to know three fundamental rules. Hover over each card to see them highlighted:
📐 The Power-Root Rule
$$(\sqrt[n]{a})^m = \sqrt[n]{a^m}$$Example: $(\sqrt{2})^5 = \sqrt{2^5}$
💡 The power can move inside or outside the radical
🔢 The Rational Exponent Rule
$$\sqrt[n]{a^m} = a^{\frac{m}{n}}$$Example: $\sqrt{2^5} = 2^{\frac{5}{2}}$
💡 The root index becomes the denominator of the exponent
➖ The Negative Exponent Rule
$$\frac{1}{a^n} = a^{-n}$$Example: $\frac{1}{2^{5/2}} = 2^{-\frac{5}{2}}$
💡 Moving from denominator to numerator flips the sign of the exponent
2. Step-by-Step Equivalence
The expression $\frac{1}{(\sqrt{2})^5}$ can be written in several mathematically identical ways. This is often a "Select All That Apply" style question in algebra.
| Form | Mathematical Expression | Why it's the same |
|---|---|---|
| Radical Power | $\frac{1}{\sqrt{2^5}}$ | The power moves inside the square root. |
| Simplified Base | $\frac{1}{\sqrt{32}}$ | Calculated $2 \times 2 \times 2 \times 2 \times 2 = 32$. |
| Fractional Exponent | $\frac{1}{2^{5/2}}$ | The square root becomes a denominator of 2. |
| Negative Exponent | $2^{-\frac{5}{2}}$ | Moving the base to the top makes the exponent negative. |
| Rationalized | $\frac{\sqrt{2}}{8}$ | Multiplied top and bottom by $\sqrt{2}$ to remove the root. |
🎯 Interactive Transformation
Click here to see the step-by-step transformation of $\frac{1}{(\sqrt{2})^5}$ → $2^{-\frac{5}{2}}$
Step 1: Apply Power-Root Rule
$$\frac{1}{(\sqrt{2})^5} = \frac{1}{\sqrt{2^5}}$$
Step 2: Apply Rational Exponent Rule
$$\frac{1}{\sqrt{2^5}} = \frac{1}{2^{5/2}}$$
Step 3: Apply Negative Exponent Rule
$$\frac{1}{2^{5/2}} = 2^{-\frac{5}{2}}$$
✅ Final Answer: $2^{-\frac{5}{2}}$
🧪 Try It Yourself!
Transform $\frac{1}{(\sqrt{5})^3}$ into exponential form with a negative exponent.
Solution:
1. Apply Power-Root Rule: $\frac{1}{\sqrt{5^3}}$
2. Apply Rational Exponent Rule: $\frac{1}{5^{3/2}}$
3. Apply Negative Exponent Rule: $5^{-\frac{3}{2}}$
Answer: $5^{-\frac{3}{2}}$
3. Common Traps to Avoid
- ❌ Sign Errors: Unless there is a minus sign in front of the entire original fraction, the result will always be positive. $-2^{\frac{5}{2}}$ is not the same as $2^{-\frac{5}{2}}$.
- ❌ Reciprocal Confusion: A negative exponent ($x^{-2}$) means "flip the fraction," not "make the number negative."
- ❌ Root vs. Multiplier: $\sqrt[2]{x}$ is not the same as $\frac{1}{2}x$. The root is an exponent ($x^{1/2}$).
4. Quick Practice
Problem 1: Which of these are equal to $\frac{1}{(\sqrt{3})^4}$?
Click the button below to reveal the answers with explanations!
- ✓ A: $\frac{1}{9}$ — Correct! Because $(\sqrt{3})^4 = (3^{1/2})^4 = 3^2 = 9$
- ✓ B: $3^{-2}$ — Correct! Because $\frac{1}{3^2} = 3^{-2} = \frac{1}{9}$
- ✗ C: $-3^2$ — Incorrect! This equals $-9$, not $\frac{1}{9}$. The negative sign is outside!
- ✓ D: $3^{-4/2}$ — Correct! Simplifies to $3^{-2} = \frac{1}{9}$
🧪 Problem 2: Simplify $\sqrt[3]{8^2}$
Convert to exponential form and simplify.
Step-by-step solution:
1. Apply Rational Exponent Rule: $\sqrt[3]{8^2} = 8^{2/3}$
2. Recognize that $8 = 2^3$: $(2^3)^{2/3}$
3. Multiply exponents: $2^{3 \cdot 2/3} = 2^2$
Answer: $4$ (or $2^2$)
🧪 Problem 3: Rewrite $x^{-3/4}$ as a radical
Express using radical notation instead of fractional exponents.
Step-by-step solution:
1. The negative exponent means it's in the denominator: $\frac{1}{x^{3/4}}$
2. The denominator 4 means fourth root: $\frac{1}{\sqrt[4]{x^3}}$
Answer: $\frac{1}{\sqrt[4]{x^3}}$
Alternative form: $\frac{1}{\sqrt[4]{x \cdot x \cdot x}}$
🌟 Bonus Challenge
Click to reveal: Simplify $\frac{(\sqrt{x})^3}{x^{-1/2}}$
Solution:
1. Convert to exponential form: $\frac{(x^{1/2})^3}{x^{-1/2}}$
2. Simplify numerator: $\frac{x^{3/2}}{x^{-1/2}}$
3. Subtract exponents (division rule): $x^{3/2 - (-1/2)} = x^{3/2 + 1/2}$
4. Add the exponents: $x^{4/2} = x^2$
Answer: $x^2$