Expression Comparison: Finding the Highest Value Among Mathematical Expressions

Q: Can I compare exponents without calculating?

No! The size of the exponent doesn't tell you which expression has the highest value. For example, $ 5^6 $ has a bigger exponent than $ 12^4 $, but $ 12^4 = 20,736 $ is larger than $ 5^6 = 15,625 $.

Q: Is there a shortcut to avoid all these calculations?

Not really for this type of problem! You need the actual values to compare. However, you can use a calculator to speed up the arithmetic, or recognize patterns like $ 12^2 = 144 $ to build up to $ 12^4 $.

Q: What if I make a calculation error?

Double-check your work by breaking down each calculation step by step. For $ 12^4 $, first find $ 12^2 = 144 $, then $ 144^2 = 20,736 $. This method helps catch mistakes!

Q: Why is 12⁴ bigger than 5⁶ if 5⁶ has more multiplication?

Because the base matters more than the exponent count! Even though you multiply 5 six times, you're multiplying a smaller number. Meanwhile, 12 multiplied four times gives a much larger result because 12 is more than twice as big as 5.

Q: How do I remember the order of these calculations?

$ 3^5 = 243 $ (smallest)$ 9^4 = 6,561 $$ 5^6 = 15,625 $$ 12^4 = 20,736 $ (largest)Notice that the larger bases generally give larger results, even with smaller exponents!

Exponent Calculations with Value Comparison

Which of the expressions below has the highest value?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Choose the largest expression

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

00:18 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Which of the expressions below has the highest value?

Step-by-step solution

To solve this problem, we shall compute the value of each expression:

First, calculate $12^4$ :
$12^4 = 12 \times 12 \times 12 \times 12 = 20736$ .
Next, calculate $9^4$ :
$9^4 = 9 \times 9 \times 9 \times 9 = 6561$ .
Then, calculate $5^6$ :
$5^6 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 15625$ .
Finally, calculate $3^5$ :
$3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243$ .

After computing the values, we compare them:

$12^4 = 20736$
$9^4 = 6561$
$5^6 = 15625$
$3^5 = 243$

Among these, the highest value is $20736$ , which corresponds to $12^4$ .

Therefore, the expression with the highest value is $12^4$ .

Final Answer

$12^4$

Key Points to Remember

Essential concepts to master this topic

Rule: Calculate each exponent by multiplying the base repeatedly
Technique: Break down step by step: 12⁴ = 12 × 12 × 12 × 12 = 20,736
Check: Compare final numerical values to find the largest: 20,736 > 15,625 > 6,561 > 243 ✓

Common Mistakes

Avoid these frequent errors

Comparing exponents instead of actual values
Don't assume 5⁶ is largest because 6 is the highest exponent = wrong comparison! Different bases make exponent size meaningless for value comparison. Always calculate the actual numerical value of each expression first.

Practice Quiz

Test your knowledge with interactive questions

Which of the following is equivalent to the expression below?

$$ $$ 10,000^1 $$

FAQ

Everything you need to know about this question

Can I compare exponents without calculating?

No! The size of the exponent doesn't tell you which expression has the highest value. For example, $5^6$ has a bigger exponent than $12^4$ , but $12^4 = 20,736$ is larger than $5^6 = 15,625$ .

Is there a shortcut to avoid all these calculations?

Not really for this type of problem! You need the actual values to compare. However, you can use a calculator to speed up the arithmetic, or recognize patterns like $12^2 = 144$ to build up to $12^4$ .

What if I make a calculation error?

Double-check your work by breaking down each calculation step by step. For $12^4$ , first find $12^2 = 144$ , then $144^2 = 20,736$ . This method helps catch mistakes!

Why is 12⁴ bigger than 5⁶ if 5⁶ has more multiplication?

Because the base matters more than the exponent count! Even though you multiply 5 six times, you're multiplying a smaller number. Meanwhile, 12 multiplied four times gives a much larger result because 12 is more than twice as big as 5.

How do I remember the order of these calculations?

$3^5 = 243$ (smallest)
$9^4 = 6,561$
$5^6 = 15,625$
$12^4 = 20,736$ (largest)

Notice that the larger bases generally give larger results, even with smaller exponents!

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