Multiply Powers with Base 5: Solving 5^(-8) × 5^6

Q: Why do we add exponents when multiplying powers?

Because exponents represent repeated multiplication! $ 5^{-8} \times 5^6 $ means we have 5 multiplied by itself -8 times, then 6 more times, for a total of -8 + 6 = -2 times.

Q: What does a negative exponent actually mean?

A negative exponent means "one divided by the positive power". So $ 5^{-2} = \frac{1}{5^2} = \frac{1}{25} $. It's the reciprocal of the positive power!

Q: How can both answers a and b be correct?

Because $ 5^{-2} $ and $ \frac{1}{5^2} $ are exactly the same value written in different forms! Negative exponents and fractions are just two ways to express the same mathematical relationship.

Q: What if the exponents were both negative?

You still add them! For example: $ 5^{-3} \times 5^{-4} = 5^{-3+(-4)} = 5^{-7} $. Remember that adding a negative number is the same as subtracting.

Q: Can I check my answer by calculating the actual value?

Absolutely! $ 5^{-8} = \frac{1}{390625} $ and $ 5^6 = 15625 $, so $ \frac{1}{390625} \times 15625 = \frac{1}{25} $, which equals $ 5^{-2} $ ✓

Exponent Rules with Negative Powers

Insert the corresponding expression:

$5^{-8}\times5^6=$

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

00:06 equals the same base raised to the sum of the exponents (N+M)

00:09 We will apply this formula to our exercise

00:13 We'll maintain the base and add the exponents together

00:20 According to the laws of exponents, any number with a negative exponent (-N)

00:24 equals its reciprocal raised to the opposite exponent (N)

00:27 We will apply this formula to our exercise

00:31 We'll convert it to the reciprocal and raise it to the opposite power

00:35 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$5^{-8}\times5^6=$

Step-by-step solution

Let's simplify the expression $5^{-8} \times 5^6$ using the rules of exponents.

Step 1: Apply the rule for multiplying powers with the same base. According to this rule, when multiplying like bases, we add the exponents: $5^{-8} \times 5^6 = 5^{-8 + 6}$
Step 2: Calculate the sum of the exponents: $-8 + 6 = -2$ .
Step 3: Write the simplified expression: $5^{-2}$ .
Step 4: Relate the expression to the given choices:

The simplified expression $5^{-2}$ corresponds to choice 1. Additionally, rewriting a negative exponent using the fraction format gives: $5^{-2} = \frac{1}{5^2}$ , which matches choice 2.

Thus, both choices 'a: $5^{-2}$ ' and 'b: $\frac{1}{5^2}$ ' are correct.

Therefore, according to the given answer choice, a'+b' are correct.

Final Answer

a'+b' are correct

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add the exponents
Technique: $5^{-8} \times 5^6 = 5^{-8+6} = 5^{-2}$
Check: Verify $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply -8 × 6 = -48 to get $5^{-48}$ ! This confuses the multiplication rule with the power rule. Always add exponents when multiplying powers with the same base: -8 + 6 = -2.

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?

Because exponents represent repeated multiplication! $5^{-8} \times 5^6$ means we have 5 multiplied by itself -8 times, then 6 more times, for a total of -8 + 6 = -2 times.

What does a negative exponent actually mean?

A negative exponent means "one divided by the positive power". So $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$ . It's the reciprocal of the positive power!

How can both answers a and b be correct?

Because $5^{-2}$ and $\frac{1}{5^2}$ are exactly the same value written in different forms! Negative exponents and fractions are just two ways to express the same mathematical relationship.

What if the exponents were both negative?

You still add them! For example: $5^{-3} \times 5^{-4} = 5^{-3+(-4)} = 5^{-7}$ . Remember that adding a negative number is the same as subtracting.

Can I check my answer by calculating the actual value?

Absolutely! $5^{-8} = \frac{1}{390625}$ and $5^6 = 15625$ , so $\frac{1}{390625} \times 15625 = \frac{1}{25}$ , which equals $5^{-2}$ ✓

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