Simplify the Expression: 7³ × 5² × 7⁴ × 5³ Using Laws of Exponents

Q: Why do I add the exponents instead of multiplying them?

The product rule says $ a^m \times a^n = a^{m+n} $. Think of it this way: $ 7^3 \times 7^4 $ means 7×7×7 times 7×7×7×7, which gives you 7 multiplied by itself 7 times total!

Q: What if the bases are different like 7 and 5?

When bases are different, you can't combine them using exponent rules. Keep them separate: $ 7^3 \times 5^2 $ stays as $ 7^3 \times 5^2 $.

Q: How do I organize expressions with multiple bases?

Group like bases together first! Rearrange $ 7^3 \times 5^2 \times 7^4 \times 5^3 $ to $ 7^3 \times 7^4 \times 5^2 \times 5^3 $, then apply the product rule to each group.

Q: Can I simplify this to one number?

You could calculate $ 7^7 \times 5^5 $ to get a huge number, but usually the simplified form with exponents is the preferred answer unless specifically asked to evaluate.

Q: What if I have three or more of the same base?

Same rule applies! $ 7^2 \times 7^3 \times 7^5 = 7^{2+3+5} = 7^{10} $. Just add all the exponents for that base together.

Exponent Rules with Multiple Bases

Simplify the following equation:

$7^3\times5^2\times7^4\times5^3=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:03 We'll use the commutative law and arrange equal bases together

00:13 We'll use the formula for multiplying exponents

00:16 Any number (A) to the power of (M) times the same number (A) to the power of (N)

00:19 We get the same number (A) to the power of the sum of exponents (M+N)

00:22 We'll use this formula in our exercise

00:25 And we'll equate the numbers to the variables in the formula

00:31 We'll keep the base and add the exponents

00:55 We'll use the same method for the second base

01:15 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

Simplify the following equation:

$7^3\times5^2\times7^4\times5^3=$

Step-by-step solution

To solve this problem, we'll use the product of powers property which states $a^m \times a^n = a^{m+n}$ .

Step 1: Simplify the expression by grouping the like bases. The original expression is $7^3 \times 5^2 \times 7^4 \times 5^3$ .
Step 2: Combine the exponents for each base. For base 7: $7^3 \times 7^4 = 7^{3+4} = 7^7$ . For base 5: $5^2 \times 5^3 = 5^{2+3} = 5^5$ .
Step 3: Write the simplified expression. After combining the exponents, the expression becomes $7^7 \times 5^5$ .

Thus, the solution to the problem is $7^7 \times 5^5$ .

Final Answer

$7^7\times5^5$

Key Points to Remember

Essential concepts to master this topic

Product Rule: When multiplying same bases, add the exponents together
Technique: Group like bases: $7^3 \times 7^4 = 7^{3+4} = 7^7$
Check: Count exponents for each base: 7 appears twice, 5 appears twice ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply the exponents like 7³ × 7⁴ = 7¹² = wrong answer! This confuses the product rule with the power rule. Always add exponents when multiplying same bases: 7³ × 7⁴ = 7³⁺⁴ = 7⁷.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I add the exponents instead of multiplying them?

The product rule says $a^m \times a^n = a^{m+n}$ . Think of it this way: $7^3 \times 7^4$ means 7×7×7 times 7×7×7×7, which gives you 7 multiplied by itself 7 times total!

What if the bases are different like 7 and 5?

When bases are different, you can't combine them using exponent rules. Keep them separate: $7^3 \times 5^2$ stays as $7^3 \times 5^2$ .

How do I organize expressions with multiple bases?

Group like bases together first! Rearrange $7^3 \times 5^2 \times 7^4 \times 5^3$ to $7^3 \times 7^4 \times 5^2 \times 5^3$ , then apply the product rule to each group.

Can I simplify this to one number?

You could calculate $7^7 \times 5^5$ to get a huge number, but usually the simplified form with exponents is the preferred answer unless specifically asked to evaluate.

What if I have three or more of the same base?

Same rule applies! $7^2 \times 7^3 \times 7^5 = 7^{2+3+5} = 7^{10}$ . Just add all the exponents for that base together.

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