Find the Missing Exponent: Solving 32 = (1/2)^x

Q: Why is the answer negative when we're looking for a positive result?

The negative exponent is what makes $ \left(\frac{1}{2}\right)^{-5} $ equal to a number greater than 1! When you have a negative exponent with a fraction, it flips the fraction and makes the exponent positive.

Q: How do I remember that 32 equals 2 to what power?

Start with powers of 2: $ 2^1=2, 2^2=4, 2^3=8, 2^4=16, 2^5=32 $. Practice these basic powers - they come up frequently in exponential problems!

Q: What if I can't see the pattern with powers of 2?

Try working backwards! Since $ \left(\frac{1}{2}\right)^x $ means dividing by 2 repeatedly, ask yourself: How many times do I multiply by 2 to get from 1 to 32? That gives you the magnitude of your exponent.

Q: Can I solve this using logarithms instead?

Yes! You can use $ \log_{1/2}(32) = x $, but converting to the same base method is often simpler and doesn't require a calculator.

Q: Why does a negative exponent make the answer bigger?

With fractions, negative exponents flip the fraction! So $ \left(\frac{1}{2}\right)^{-5} = \left(\frac{2}{1}\right)^5 = 2^5 = 32 $. The negative sign transforms the fraction into its reciprocal.

Exponential Equations with Negative Exponents

$32=(\frac{1}{2})^?$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Insert the missing exponent

00:03 Let's break down 32 to 2 to the power of 5

00:09 In order to eliminate a negative exponent

00:13 We'll flip the numerator and denominator and the exponent will become positive

00:18 Apply this formula to our exercise

00:28 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$32=(\frac{1}{2})^?$

Step-by-step solution

To solve this problem, let's follow these steps:

Step 1: Recognize that 32 can be expressed as a power of 2.
Step 2: Use negative exponents to equate $\left(\frac{1}{2}\right)^x$ to this power of 2.
Step 3: Solve for the unknown exponent $x$ .

Now, let's work through each step:
Step 1: The number 32 can be expressed as a power of 2. Specifically, $32 = 2^5$ .
Step 2: The expression $\left(\frac{1}{2}\right)^x$ is equivalent to $\left(2^{-1}\right)^x = 2^{-x}$ . To find $x$ , equate: $2^5 = 2^{-x}$ .
Step 3: Since the bases are equal, the exponents must be equal. Therefore, we have $5 = -x$ . Solving for $x$ , we multiply both sides by -1 and get $x = -5$ .

Therefore, the solution to the problem is $x = -5$ .

Final Answer

$-5$

Key Points to Remember

Essential concepts to master this topic

Rule: Convert fractions to negative exponents: $\left(\frac{1}{2}\right)^x = 2^{-x}$
Technique: Express both sides as powers of same base: $32 = 2^5$ and $2^{-x}$
Check: Verify: $\left(\frac{1}{2}\right)^{-5} = 2^5 = 32$ ✓

Common Mistakes

Avoid these frequent errors

Making the exponent positive instead of negative
Don't think $\left(\frac{1}{2}\right)^5 = 32$ = $\frac{1}{32}$ ! This gives the reciprocal of what you need. Always remember that $\left(\frac{1}{a}\right)^x = a^{-x}$ , so negative exponents flip the fraction.

Practice Quiz

Test your knowledge with interactive questions

$$ (2^3)^6 = $$

FAQ

Everything you need to know about this question

Why is the answer negative when we're looking for a positive result?

The negative exponent is what makes $\left(\frac{1}{2}\right)^{-5}$ equal to a number greater than 1! When you have a negative exponent with a fraction, it flips the fraction and makes the exponent positive.

How do I remember that 32 equals 2 to what power?

Start with powers of 2: $2^1=2, 2^2=4, 2^3=8, 2^4=16, 2^5=32$ . Practice these basic powers - they come up frequently in exponential problems!

What if I can't see the pattern with powers of 2?

Try working backwards! Since $\left(\frac{1}{2}\right)^x$ means dividing by 2 repeatedly, ask yourself: How many times do I multiply by 2 to get from 1 to 32? That gives you the magnitude of your exponent.

Can I solve this using logarithms instead?

Yes! You can use $\log_{1/2}(32) = x$ , but converting to the same base method is often simpler and doesn't require a calculator.

Why does a negative exponent make the answer bigger?

With fractions, negative exponents flip the fraction! So $\left(\frac{1}{2}\right)^{-5} = \left(\frac{2}{1}\right)^5 = 2^5 = 32$ . The negative sign transforms the fraction into its reciprocal.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Exponents Rules questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

Find the Missing Exponent: Solving 32 = (1/2)^x

Exponential Equations with Negative 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 is the answer negative when we're looking for a positive result?

How do I remember that 32 equals 2 to what power?

What if I can't see the pattern with powers of 2?

Can I solve this using logarithms instead?

Why does a negative exponent make the answer bigger?

🌟 Unlock Your Math Potential

More Questions

Exponents Rules

Solve: 7^(-8) × (1/7)^(5x) = 49 | Exponent Equation Challenge

Solve the Exponential Equation: (1/5)² Raised to Unknown Power Divided by 5 Equals 125

Simplify Complex Expression: (x^4⋅x^8⋅x^(-3))/(x^(-4)) ÷ (x^10⋅x^3)/x^5

Solve (-4/9)^-3: Negative Exponent with Fractions

Negative Exponents

Solve the Exponential Equation: 4^(x+8) = 1/256

Solve for the Unknown Exponent: 9^x(1/2)^-4 = 16/3

Solve for the Missing Exponent: 4^x = 1/64 Equation

Solve 7^-4 Minus 7^-8: Negative Exponent Subtraction Problem

Powers of a Fraction

Solve: 5⁴ × (1/5)⁴ - Multiplying Powers with Same Base

Simplify This Complex Expression: (13/2)^0 ⋅ (2/13)^(-2) ⋅ (13/2)^(-5)

Solve: (a/b)^(-7) · b^8 = ? · b^(-4) · (b/a)^7

Solve (7/8)^(-4) · 8/7 + 7/8 · (8/7)^(-3): Complete Fraction Expression

Applying Combined Exponents Rules

Simplify 7^x × 7^(-x): Multiplying Exponential Expressions

Examine the Power of Inversion: Simplify 300^-4 x (1/300)^-4

Simplify the Complex Fraction: Dividing Powers and Managing Negative Exponents

Simplify and Solve: 2^2·2^-3·2^4 / 2^3·2^-2·2^5