Solve: [(1/7)^-1]^4 Using Laws of Negative Exponents

Q: Why does $ (\frac{1}{7})^{-1} $ equal 7?

The negative exponent rule says $ a^{-1} = \frac{1}{a} $. So $ (\frac{1}{7})^{-1} = \frac{1}{\frac{1}{7}} = 7 $. Think of it as flipping the fraction!

Q: How do I handle multiple layers of exponents?

Work from the innermost parentheses outward. First simplify $ (\frac{1}{7})^{-1} = 7 $, then apply the outer exponent: $ 7^4 $.

Q: When do I multiply exponents vs. add them?

Multiply when you have a power raised to another power: $ (a^m)^n = a^{m \cdot n} $. Add when multiplying same bases: $ a^m \cdot a^n = a^{m+n} $.

Q: What if I get confused with all the negative signs?

Take it step by step! Write each transformation clearly: $ \frac{1}{7} = 7^{-1} $, then $ (7^{-1})^{-1} = 7^{(-1) \times (-1)} = 7^1 $, finally $ (7^1)^4 = 7^4 $.

Q: How can I check if $ 7^4 $ is really the answer?

Calculate both sides! $ 7^4 = 2401 $ and the original expression $ [(\frac{1}{7})^{-1}]^4 = 7^4 = 2401 $. When they match, you're correct!

Power Rules with Nested Exponents

$[(\frac{1}{7})^{-1}]^4=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:06 We'll use the formula for power of power

00:10 Any number (A) raised to power (N) raised to power (M)

00:13 equals the same number (A) raised to the power of the sum of exponents (M+N)

00:16 We'll use this formula in our exercise

00:21 We'll multiply between the powers and calculate

00:29 Now we'll use the formula for negative exponent

00:32 When we have a negative exponent (-M) on any fraction (A/B)

00:36 We get the reciprocal number (B/A) raised to the positive exponent (M)

00:46 We'll use this formula in our exercise

00:50 We'll substitute the reciprocal number and the opposite exponent

00:53 And this is the solution to the question

01:03 New Chapter

Step-by-step written solution

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Understand the problem

$[(\frac{1}{7})^{-1}]^4=$

Step-by-step solution

We use the power property of a negative exponent:

$a^{-n}=\frac{1}{a^n}$ We will rewrite the fraction in parentheses as a negative power:

$\frac{1}{7}=7^{-1}$ Let's return to the problem, where we had:

$\bigg( \big( \frac{1}{7}\big)^{-1}\bigg)^4=\big((7^{-1})^{-1} \big)^4$ We continue and use the power property of an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$ And we apply it in the problem:

$\big((7^{-1})^{-1} \big)^4 =(7^{-1\cdot-1})^4=(7^1)^4=7^{1\cdot4}=7^4$ Therefore, the correct answer is option c

Final Answer

$7^4$

Key Points to Remember

Essential concepts to master this topic

Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$ flips base to reciprocal
Power of Power Rule: $(a^m)^n = a^{m \cdot n}$ so $(7^{-1})^{-1} = 7^1$
Check Final Result: $[( \frac{1}{7})^{-1}]^4 = 7^4 = 2401$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying them
Don't add exponents like (-1) + (-1) + 4 = 2 giving $7^2$ ! Power rules require multiplication, not addition. Always multiply exponents: (-1) × (-1) × 4 = 4 giving $7^4$ .

Practice Quiz

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$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does $(\frac{1}{7})^{-1}$ equal 7?

The negative exponent rule says $a^{-1} = \frac{1}{a}$ . So $(\frac{1}{7})^{-1} = \frac{1}{\frac{1}{7}} = 7$ . Think of it as flipping the fraction!

How do I handle multiple layers of exponents?

Work from the innermost parentheses outward. First simplify $(\frac{1}{7})^{-1} = 7$ , then apply the outer exponent: $7^4$ .

When do I multiply exponents vs. add them?

Multiply when you have a power raised to another power: $(a^m)^n = a^{m \cdot n}$ . Add when multiplying same bases: $a^m \cdot a^n = a^{m+n}$ .

What if I get confused with all the negative signs?

Take it step by step! Write each transformation clearly: $\frac{1}{7} = 7^{-1}$ , then $(7^{-1})^{-1} = 7^{(-1) \times (-1)} = 7^1$ , finally $(7^1)^4 = 7^4$ .

How can I check if $7^4$ is really the answer?

Calculate both sides! $7^4 = 2401$ and the original expression $[(\frac{1}{7})^{-1}]^4 = 7^4 = 2401$ . When they match, you're correct!

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Solve: [(1/7)^-1]^4 Using Laws of Negative Exponents

Power Rules with Nested Exponents

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why does $(\frac{1}{7})^{-1}$ equal 7?

How do I handle multiple layers of exponents?

When do I multiply exponents vs. add them?

What if I get confused with all the negative signs?

How can I check if $7^4$ is really the answer?

🌟 Unlock Your Math Potential

More Questions

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Simplify: 4^(2y) × 4^(-5) × 4^(-y) × 4^6 Using Laws of Exponents

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Negative Exponents

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Simplify x^-a: Understanding Negative Exponent Notation

Simplify 5^-2: Negative Exponent Calculation Step-by-Step

Evaluate (1/4)^(-1): Converting Negative Exponents to Reciprocals

Power of a Power

Solve: (2²)³ + (3³)⁴ + (9²)⁶ - Nested Exponents Challenge

Solve ((7×3)²)⁶ + (3⁻¹)³×(2³)⁴: Complex Exponent Challenge

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Solve (3^5)^4: Evaluating Nested Exponent Expression

Solve: [(1/7)^-1]^4 Using Laws of Negative Exponents

Power Rules with Nested Exponents

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why does (17)−1 (\frac{1}{7})^{-1} (71​)−1 equal 7?

How do I handle multiple layers of exponents?

When do I multiply exponents vs. add them?

What if I get confused with all the negative signs?

How can I check if 74 7^4 74 is really the answer?

🌟 Unlock Your Math Potential

More Questions

Applying Combined Exponents Rules

Simplify: 4^(2y) × 4^(-5) × 4^(-y) × 4^6 Using Laws of Exponents

Simplify the Complex Fraction: 1/(X^7/X^6) Step by Step

Solve (4²/7⁴)²: Calculating Powers of Complex Fractions

Calculate (2/6)³: Cube of a Simple Fraction

Exponents Rules

Simplify Base-7 Expression: 7^(2x+1) × 7^(-1) × 7^x

Calculate 2^(-5): Solving Negative Exponent Expression

Simplify the Expression: 2^(2x+1) · 2^5 · 2^(3x)

Evaluate 5⁰: Understanding the Zero Exponent Rule

Negative Exponents

Solve for 4^(-1): Converting Negative Exponent to Reciprocal

Simplify x^-a: Understanding Negative Exponent Notation

Simplify 5^-2: Negative Exponent Calculation Step-by-Step

Evaluate (1/4)^(-1): Converting Negative Exponents to Reciprocals

Power of a Power

Solve: (2²)³ + (3³)⁴ + (9²)⁶ - Nested Exponents Challenge

Solve ((7×3)²)⁶ + (3⁻¹)³×(2³)⁴: Complex Exponent Challenge

Solve (6²)¹³: Calculating a Power of Power Expression

Solve (3^5)^4: Evaluating Nested Exponent Expression

Why does $(\frac{1}{7})^{-1}$ equal 7?

How can I check if $7^4$ is really the answer?