Expand 7^6: Calculate the Sixth Power of Seven

Q: Why do we add exponents when multiplying powers with the same base?

When you multiply $ 7^2 \times 7^2 $, you're really doing (7×7) × (7×7), which gives you four 7's multiplied together = $ 7^4 $. The exponents tell you how many times to use the base!

Q: How can I remember whether to add or multiply exponents?

Multiplication of powers: Add exponents ($ a^m \times a^n = a^{m+n} $)Power of a power: Multiply exponents ($ (a^m)^n = a^{m \times n} $)Think: multiply the bases, add the powers!

Q: What does it mean when the problem says 'b+c are correct'?

This means multiple answer choices are right! You need to check each option and see which ones actually equal $ 7^6 $. In this case, both choice b and choice c simplify to the same value.

Q: Do I need to calculate the actual value of 7^6?

Not necessarily! You can work with the exponent rules to see which expressions are equivalent. However, calculating $ 7^6 = 117,649 $ can help you double-check your work.

Q: What if I see 7 without an exponent in the expression?

Remember that 7 = 7^1! Any number without a visible exponent has an implied exponent of 1. So $ 7^1 \times 7 \times 7^4 $ becomes $ 7^1 \times 7^1 \times 7^4 = 7^6 $.

Exponent Rules with Multiple Choice Validation

Expand the following expression:

$7^6=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Identify the equivalent expressions

00:03 According to the power laws, the multiplication of powers with an equal base (A)

00:08 equals the same base raised to the sum of the exponents (N+M)

00:12 We will apply this formula to our exercise in order to simplify it

00:16 We'll maintain the base and add the exponents together

00:20 We can observe that this expression is not equal to the original expression

00:26 Let's simplify the remaining expressions in the same way

00:38 This expression is equal to the original expression

00:49 And this expression is also equal to the original expression

00:54 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Expand the following expression:

$7^6=$

Step-by-step solution

To solve this problem, let's examine the possible answer choices to determine which ones equal $7^6$ .

**Choice 1:** $7^1 \times 7^2 \times 7^4$
By exponent rules: $7^1 \cdot 7^2 \cdot 7^4 = 7^{1+2+4} = 7^7$ .
**Choice 2:** $7^1 \times 7 \times 7^4$
Here, $7 = 7^1$ . So, $7^1 \cdot 7^1 \cdot 7^4 = 7^{1+1+4} = 7^6$ .
**Choice 3:** $7^2 \times 7^2 \times 7^2$
Using the rule: $7^2 \cdot 7^2 \cdot 7^2 = 7^{2+2+2} = 7^6$ .
**Choice 4:** This states choices 'b + c are correct'.

After calculations, choices 2 and 3 simplify to $7^6$ . Therefore, the correct answer is indeed that choices 'b+c are correct'. Thus, the correct choice is:

Choice 4: b+c are correct

Final Answer

b+c are correct

Key Points to Remember

Essential concepts to master this topic

Exponent Rule: When multiplying same bases, add the exponents together
Technique: $7^2 \times 7^2 \times 7^2 = 7^{2+2+2} = 7^6$
Check: Calculate each expression to verify which equal $7^6 = 117,649$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply exponents when bases are the same like $7^2 \times 7^2 = 7^{2 \times 2} = 7^4$ = wrong answer! This confuses the power rule with the product rule. Always add exponents when multiplying same bases: $7^2 \times 7^2 = 7^{2+2} = 7^4$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we add exponents when multiplying powers with the same base?

When you multiply $7^2 \times 7^2$ , you're really doing (7×7) × (7×7), which gives you four 7's multiplied together = $7^4$ . The exponents tell you how many times to use the base!

How can I remember whether to add or multiply exponents?

Multiplication of powers: Add exponents ( $a^m \times a^n = a^{m+n}$ )
Power of a power: Multiply exponents ( $(a^m)^n = a^{m \times n}$ )
Think: multiply the bases, add the powers!

What does it mean when the problem says 'b+c are correct'?

This means multiple answer choices are right! You need to check each option and see which ones actually equal $7^6$ . In this case, both choice b and choice c simplify to the same value.

Do I need to calculate the actual value of 7^6?

Not necessarily! You can work with the exponent rules to see which expressions are equivalent. However, calculating $7^6 = 117,649$ can help you double-check your work.

What if I see 7 without an exponent in the expression?

Remember that 7 = 7^1! Any number without a visible exponent has an implied exponent of 1. So $7^1 \times 7 \times 7^4$ becomes $7^1 \times 7^1 \times 7^4 = 7^6$ .

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