Compare Powers: Finding the Largest Value Among 7⁴, 8³, 10², and 11^(1/2)

Q: Why isn't 10² the largest since 10 is the biggest base?

The base size alone doesn't determine the final value! Even though 10 is the largest base, $ 10^2 = 100 $ while $ 7^4 = 2401 $. The combination of base and exponent matters most.

Q: What does 11^(1/2) mean exactly?

$ 11^{\frac{1}{2}} $ means the square root of 11. Any number raised to the power of $ \frac{1}{2} $ equals its square root, so $ 11^{\frac{1}{2}} = \sqrt{11} \approx 3.32 $.

Q: How do I calculate 7⁴ without a calculator?

Break it down step by step: $ 7^4 = 7 \times 7 \times 7 \times 7 $. First find $ 7^2 = 49 $, then $ 49^2 = 2401 $. Or calculate: $ 49 \times 7 = 343 $, then $ 343 \times 7 = 2401 $.

Q: Can I estimate square roots quickly?

Yes! For $ \sqrt{11} $, note that $ 3^2 = 9 $ and $ 4^2 = 16 $. Since 11 is between 9 and 16, $ \sqrt{11} $ is between 3 and 4, closer to 3.

Q: Should I always convert everything to decimals?

Not always! For comparing, you can often work with exact values when possible. But when you have mixed types like fractional exponents, converting to decimals helps make clear comparisons.

Q: What if two values are really close?

Calculate more precisely! Use a calculator or estimate more carefully. In this problem, the values are quite different (2401 vs 512), but sometimes you need exact calculations for close comparisons.

Exponent Evaluation with Mixed Power Types

Which of the expressions below has the highest value?

$7^4,8^3,10^2,11^{\frac{1}{2}}$

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Step-by-step video solution

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00:00 Choose the largest expression

00:03 We'll solve each exponent and choose the largest one

00:13 And this is the solution to the question

Step-by-step written solution

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Understand the problem

Which of the expressions below has the highest value?

$7^4,8^3,10^2,11^{\frac{1}{2}}$

Step-by-step solution

The goal is to compare the values of the expressions $7^4$ , $8^3$ , $10^2$ , and $11^{\frac{1}{2}}$ to identify the one with the highest value.

Let's calculate each expression:

Calculate $7^4$ :
- $7^4 = 7 \times 7 \times 7 \times 7$
- $7 \times 7 = 49$
- $49 \times 7 = 343$
- $343 \times 7 = 2401$
Therefore, $7^4 = 2401$ .
Calculate $8^3$ :
- $8^3 = 8 \times 8 \times 8$
- $8 \times 8 = 64$
- $64 \times 8 = 512$
Therefore, $8^3 = 512$ .
Calculate $10^2$ :
- $10^2 = 10 \times 10 = 100$ .
Therefore, $10^2 = 100$ .
Calculate $11^{\frac{1}{2}}$ (the square root of 11):
- $11^{\frac{1}{2}} \approx \sqrt{11} \approx 3.3166$ .
Therefore, $11^{\frac{1}{2}} \approx 3.3166$ .

Now, let's compare these values:

$7^4 = 2401$
$8^3 = 512$
$10^2 = 100$
$11^{\frac{1}{2}} \approx 3.3166$

Clearly, $2401$ is the highest value among these, which is obtained from $7^4$ .

Therefore, the expression with the highest value is $7^4$ .

Final Answer

$7^4$

Key Points to Remember

Essential concepts to master this topic

Rule: Calculate each power completely before comparing values
Technique: Convert fractional exponents: $11^{\frac{1}{2}} = \sqrt{11} \approx 3.32$
Check: Compare final values: 2401 > 512 > 100 > 3.32 ✓

Common Mistakes

Avoid these frequent errors

Comparing exponents instead of actual values
Don't assume higher exponents mean larger results like 10² > 8³ because 2 < 3 = wrong comparison! Exponent size doesn't determine value when bases differ. Always calculate the actual numerical value of each expression first.

Practice Quiz

Test your knowledge with interactive questions

$$ 11^2= $$

FAQ

Everything you need to know about this question

Why isn't 10² the largest since 10 is the biggest base?

The base size alone doesn't determine the final value! Even though 10 is the largest base, $10^2 = 100$ while $7^4 = 2401$ . The combination of base and exponent matters most.

What does 11^(1/2) mean exactly?

$11^{\frac{1}{2}}$ means the square root of 11. Any number raised to the power of $\frac{1}{2}$ equals its square root, so $11^{\frac{1}{2}} = \sqrt{11} \approx 3.32$ .

How do I calculate 7⁴ without a calculator?

Break it down step by step: $7^4 = 7 \times 7 \times 7 \times 7$ . First find $7^2 = 49$ , then $49^2 = 2401$ . Or calculate: $49 \times 7 = 343$ , then $343 \times 7 = 2401$ .

Can I estimate square roots quickly?

Yes! For $\sqrt{11}$ , note that $3^2 = 9$ and $4^2 = 16$ . Since 11 is between 9 and 16, $\sqrt{11}$ is between 3 and 4, closer to 3.

Should I always convert everything to decimals?

Not always! For comparing, you can often work with exact values when possible. But when you have mixed types like fractional exponents, converting to decimals helps make clear comparisons.

What if two values are really close?

Calculate more precisely! Use a calculator or estimate more carefully. In this problem, the values are quite different (2401 vs 512), but sometimes you need exact calculations for close comparisons.

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