Simplify 7^4 × 7: Power Multiplication Problem

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

Because exponents represent repeated multiplication! $ 7^4 $ means 7×7×7×7, and $ 7^1 $ means 7. When you multiply them, you get 7×7×7×7×7, which is $ 7^5 $.

Q: What if the number doesn't have an exponent written?

Any number without a visible exponent has an invisible exponent of 1. So 7 is really $ 7^1 $, making $ 7^4 \times 7 = 7^4 \times 7^1 $.

Q: Does this rule work with different bases like 3 and 5?

No! The rule $ a^m \times a^n = a^{m+n} $ only works when the bases are identical. For different bases like $ 3^2 \times 5^3 $, you cannot combine the exponents.

Q: How can I remember this rule?

Think of it as collecting multiplication! When you multiply powers of the same number, you're just adding more copies of that number being multiplied together, so the exponents add up.

Q: What's the difference between this and the power of a power rule?

This is multiplication of powers: $ 7^4 \times 7^1 = 7^{4+1} $ (add exponents). The power of a power rule is $ (7^4)^1 = 7^{4 \times 1} $ (multiply exponents).

Exponent Rules with Same Base Multiplication

Simplify the following equation:

$7^4\times7=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 Any number raised to the power of 1 is always equal to itself

00:06 We'll apply this formula to our exercise, and raise to the power of 1

00:13 According to the laws of exponents, the multiplication of powers with the same base (A)

00:17 equals the same base raised to the sum of the exponents (N+M)

00:22 We'll apply this formula to our exercise

00:23 We'll then proceed to add up the exponents and raise them to this power

00:29 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following equation:

$7^4\times7=$

Step-by-step solution

To simplify the expression $7^4 \times 7$ , we follow these steps:

Step 1: Recognize that $7$ is equivalent to $7^1$ . Thus, our expression becomes $7^4 \times 7^1$ .
Step 2: Apply the rule of multiplying powers with the same base, which states: $a^m \times a^n = a^{m+n}$ .
Step 3: According to the rule, add the exponents of the same base: $4 + 1$ .
Step 4: Simplify the result, yielding $7^{4+1} = 7^5$ .

Thus, the simplified expression is $7^5$ .

Final Answer

$7^5$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add the exponents
Technique: Rewrite 7 as $7^1$ , then $7^4 \times 7^1 = 7^{4+1}$
Check: Verify $7^5 = 7 \times 7 \times 7 \times 7 \times 7$ equals original expression ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply 4 × 1 = 4 to get $7^4$ ! This ignores the fundamental rule for same-base multiplication and gives the wrong power. Always add exponents when multiplying powers with the same base.

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?

Because exponents represent repeated multiplication! $7^4$ means 7×7×7×7, and $7^1$ means 7. When you multiply them, you get 7×7×7×7×7, which is $7^5$ .

What if the number doesn't have an exponent written?

Any number without a visible exponent has an invisible exponent of 1. So 7 is really $7^1$ , making $7^4 \times 7 = 7^4 \times 7^1$ .

Does this rule work with different bases like 3 and 5?

No! The rule $a^m \times a^n = a^{m+n}$ only works when the bases are identical. For different bases like $3^2 \times 5^3$ , you cannot combine the exponents.

How can I remember this rule?

Think of it as collecting multiplication! When you multiply powers of the same number, you're just adding more copies of that number being multiplied together, so the exponents add up.

What's the difference between this and the power of a power rule?

This is multiplication of powers: $7^4 \times 7^1 = 7^{4+1}$ (add exponents). The power of a power rule is $(7^4)^1 = 7^{4 \times 1}$ (multiply exponents).

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