Multiply Powers: Solving 7^9 × 7^1 Using Exponent Rules

Q: What if one of the exponents is 1? Does the rule still apply?

Yes! The rule works with any exponents, including 1. Remember that $ 7^1 = 7 $, so you're just adding one more factor of 7 to your multiplication.

Q: How is this different from adding powers like 7^9 + 7^1?

Great question! Multiplication of powers uses addition of exponents, but addition of powers cannot be simplified this way. $ 7^9 + 7^1 $ stays as is - no exponent rule applies!

Q: What happens if the bases are different, like 7^9 × 3^1?

The product rule only works with the same base. If bases are different, you cannot combine the exponents. $ 7^9 \times 3^1 $ stays as $ 7^9 \times 3 $.

Q: Can I use this rule with negative or zero exponents?

Absolutely! The product rule $ a^m \times a^n = a^{m+n} $ works with any exponents - positive, negative, zero, or even fractions. Just add them together!

Exponent Rules with Same Base Multiplication

$7^9\times7^1=$

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Step-by-step written solution

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Understand the problem

$7^9\times7^1=$

Step-by-step solution

To solve the expression $7^9 \times 7^1$ , we need to apply the rules of exponents, specifically the multiplication of powers rule. According to this rule, when we multiply two powers with the same base, we keep the base and add the exponents together.

The expression can be written as $a^m \times a^n = a^{m+n}$ , where $a$ is the base and $m$ and $n$ are the exponents.
In this case, our base $a$ is 7, and our exponents are 9 and 1.
Applying the formula, we have: $7^9 \times 7^1 = 7^{9+1}$ .
Simplifying the exponent: $9 + 1 = 10$ .

Thus, the expression simplifies to: $7^{10}$ .

Final Answer

$7^{10}$

Key Points to Remember

Essential concepts to master this topic

Product Rule: When multiplying same bases, add the exponents
Technique: $7^9 \times 7^1 = 7^{9+1} = 7^{10}$
Check: Verify base stays same and exponents were added correctly ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply exponents like $7^9 \times 7^1 = 7^{9 \times 1} = 7^9$ ! This gives the wrong answer because you're applying the wrong rule. 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 we add the exponents instead of multiplying them?

When you multiply powers with the same base, you're actually adding groups of that base together. $7^9 \times 7^1$ means 9 sevens multiplied by 1 seven, giving you 10 sevens total!

What if one of the exponents is 1? Does the rule still apply?

Yes! The rule works with any exponents, including 1. Remember that $7^1 = 7$ , so you're just adding one more factor of 7 to your multiplication.

How is this different from adding powers like 7^9 + 7^1?

Great question! Multiplication of powers uses addition of exponents, but addition of powers cannot be simplified this way. $7^9 + 7^1$ stays as is - no exponent rule applies!

What happens if the bases are different, like 7^9 × 3^1?

The product rule only works with the same base. If bases are different, you cannot combine the exponents. $7^9 \times 3^1$ stays as $7^9 \times 3$ .

Can I use this rule with negative or zero exponents?

Absolutely! The product rule $a^m \times a^n = a^{m+n}$ works with any exponents - positive, negative, zero, or even fractions. Just add them together!

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