Solve the Exponential Product: 8^7 × 10^7

Q: When can I use the rule that $ a^m \times b^m = (a \times b)^m $ ?

You can use this rule only when the exponents are identical. In our problem, both 8 and 10 are raised to the power of 7, so we can combine them as $ (8 \times 10)^7 $.

Q: What if the exponents were different, like $ 8^7 \times 10^5 $ ?

If the exponents are different, you cannot use the product rule. You would need to calculate each term separately: $ 8^7 = 2,097,152 $ and $ 10^5 = 100,000 $, then multiply the results.

Q: Why is $ (4^7 \times 2^7 \times 10) $ wrong?

This option breaks down 8 into $ 4 \times 2 $ but then doesn't apply the rule correctly. It should be $ (4 \times 2 \times 10)^7 $ if we're factoring, but that's unnecessarily complicated.

Q: Do I need to calculate $ 80^7 $ to solve this?

No! The question asks for the equivalent expression, not the numerical answer. Recognizing that $ 8^7 \times 10^7 = (8 \times 10)^7 $ is the complete solution.

Q: What's the difference between this rule and $ a^m \times a^n = a^{m+n} $ ?

Great question! The rule $ a^m \times a^n = a^{m+n} $ is for same bases with different exponents. Our rule $ a^m \times b^m = (a \times b)^m $ is for different bases with the same exponent.

Exponential Properties with Product Rule

Choose the expression that corresponds to the following:

$8^7\times10^7=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following expression

00:03 When we are presented with a multiplication operation where each factor has the same exponent (N)

00:07 The entire multiplication can be written with the exponent (N)

00:12 We will apply this formula to our exercise

00:18 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the expression that corresponds to the following:

$8^7\times10^7=$

Step-by-step solution

To solve the expression $8^7 \times 10^7$ , we can use the power of a product rule, which states that $a^m \times b^m = (a \times b)^m$ . Here, $a = 8$ and $b = 10$ , and both are raised to the same power: $m = 7$ .

Following these steps:

Identify the base numbers and the common exponent: here, the base numbers are $8$ and $10$ , and the common exponent is $7$ .
Apply the power of a product rule: Instead of multiplying $8^7$ and $10^7$ directly, we apply the rule to get $(8 \times 10)^7$ .
This simplifies to $(80)^7$ .

therefore, the rewritten expression is $\left(8 \times 10\right)^7$ .

Final Answer

$\left(8\times10\right)^7$

Key Points to Remember

Essential concepts to master this topic

Rule: When bases differ but exponents match, use $a^m \times b^m = (a \times b)^m$
Technique: Convert $8^7 \times 10^7$ to $(8 \times 10)^7 = 80^7$
Check: Verify both expressions have same exponent: 7 appears on both terms ✓

Common Mistakes

Avoid these frequent errors

Trying to add exponents when bases are different
Don't change $8^7 \times 10^7$ to $(8 \times 10)^{14}$ = wrong power! Adding exponents only works when the bases are identical. Always check that both terms have the same exponent before applying the product rule.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

When can I use the rule that $a^m \times b^m = (a \times b)^m$ ?

You can use this rule only when the exponents are identical. In our problem, both 8 and 10 are raised to the power of 7, so we can combine them as $(8 \times 10)^7$ .

What if the exponents were different, like $8^7 \times 10^5$ ?

If the exponents are different, you cannot use the product rule. You would need to calculate each term separately: $8^7 = 2,097,152$ and $10^5 = 100,000$ , then multiply the results.

Why is $(4^7 \times 2^7 \times 10)$ wrong?

This option breaks down 8 into $4 \times 2$ but then doesn't apply the rule correctly. It should be $(4 \times 2 \times 10)^7$ if we're factoring, but that's unnecessarily complicated.

Do I need to calculate $80^7$ to solve this?

No! The question asks for the equivalent expression, not the numerical answer. Recognizing that $8^7 \times 10^7 = (8 \times 10)^7$ is the complete solution.

What's the difference between this rule and $a^m \times a^n = a^{m+n}$ ?

Great question! The rule $a^m \times a^n = a^{m+n}$ is for same bases with different exponents. Our rule $a^m \times b^m = (a \times b)^m$ is for different bases with the same exponent.

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Solve the Exponential Product: 8^7 × 10^7

Exponential Properties with Product Rule

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

When can I use the rule that $a^m \times b^m = (a \times b)^m$ ?

What if the exponents were different, like $8^7 \times 10^5$ ?

Why is $(4^7 \times 2^7 \times 10)$ wrong?

Do I need to calculate $80^7$ to solve this?

What's the difference between this rule and $a^m \times a^n = a^{m+n}$ ?

🌟 Unlock Your Math Potential

More Questions

Power of a Product

Solve the Power Expression: 2³ × 4³ Multiplication Problem

Calculate the Product: Solving 3^4 × 4^4 Expression

Solve the Exponential Product: 11^6 × 4^6

Calculate (12×3)^5: Evaluating the Fifth Power of a Product

Solve the Exponential Product: 8^7 × 10^7

Exponential Properties with Product Rule

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

When can I use the rule that am×bm=(a×b)m a^m \times b^m = (a \times b)^m am×bm=(a×b)m?

What if the exponents were different, like 87×105 8^7 \times 10^5 87×105?

Why is (47×27×10) (4^7 \times 2^7 \times 10) (47×27×10) wrong?

Do I need to calculate 807 80^7 807 to solve this?

What's the difference between this rule and am×an=am+n a^m \times a^n = a^{m+n} am×an=am+n?

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More Questions

Power of a Product

Solve the Power Expression: 2³ × 4³ Multiplication Problem

Calculate the Product: Solving 3^4 × 4^4 Expression

Solve the Exponential Product: 11^6 × 4^6

Calculate (12×3)^5: Evaluating the Fifth Power of a Product

When can I use the rule that $a^m \times b^m = (a \times b)^m$ ?

What if the exponents were different, like $8^7 \times 10^5$ ?

Why is $(4^7 \times 2^7 \times 10)$ wrong?

Do I need to calculate $80^7$ to solve this?

What's the difference between this rule and $a^m \times a^n = a^{m+n}$ ?