Simplify 1/((10×7)^5): Fraction with Compound Base Expression

Q: What's the difference between negative exponents and negative numbers?

A negative exponent like $ x^{-5} $ means reciprocal (flip the fraction), while a negative number like $ -x^5 $ just means opposite sign!

Q: Do I need to multiply out (10 × 7) first?

No! Keep the base as $ (10 \times 7) $ since the question asks for the expression form. The parentheses show it's treated as one unit.

Q: Why isn't the answer just 1/70^5?

While $ 10 \times 7 = 70 $, the question specifically asks to insert the corresponding expression, so we keep the format $ (10 \times 7)^{-5} $.

Q: How do I remember the negative exponent rule?

Think: "Flip it, don't negate it!" When you see $ \frac{1}{something} $, the something gets a negative exponent to move upstairs.

Q: Can I expand this to 10^(-5) × 7^(-5)?

Yes! Using the rule $ (ab)^n = a^n b^n $, we get $ (10 \times 7)^{-5} = 10^{-5} \times 7^{-5} $, but the given answer choices want the compact form.

Negative Exponents with Product Bases

Insert the corresponding expression:

$\frac{1}{\left(10\times7\right)^5}=$

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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 Apply the exponent laws in order to simplify the negative exponents

00:06 Convert to the reciprocal number and raise to the power of (-1)

00:09 We'll apply this formula to our exercise

00:12 Convert to the reciprocal number (1 divided by the number)

00:16 Proceed to raise to the power of (-1)

00:20 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:

$\frac{1}{\left(10\times7\right)^5}=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Recognize the given expression: $\frac{1}{(10 \times 7)^5}$ .
Step 2: Apply the rule of negative exponents, which states: $\frac{1}{a} = a^{-1}$ .
Step 3: Express the reciprocal with a negative exponent: $\frac{1}{(10 \times 7)^5} = (10 \times 7)^{-5}$ .

Now, let's further simplify the expression:
Given that $(a \times b)^n = a^n \times b^n$ , we can rewrite $(10 \times 7)^5$ as $10^5 \times 7^5$ . Thus, $(10 \times 7)^{-5} = (10^5 \times 7^5)^{-1} = 10^{-5} \times 7^{-5} = (10 \times 7)^{-5}$ .

Therefore, the solution to the problem in the expression form is: $\left(10 \times 7\right)^{-5}$ .

Final Answer

$\left(10\times7\right)^{-5}$

Key Points to Remember

Essential concepts to master this topic

Rule: Reciprocal becomes negative exponent: $\frac{1}{a^n} = a^{-n}$
Technique: Keep the base unchanged: $\frac{1}{(10 \times 7)^5} = (10 \times 7)^{-5}$
Check: Negative exponent means reciprocal, so answer equals original expression ✓

Common Mistakes

Avoid these frequent errors

Adding negative sign to the front of expression
Don't write $-(10 \times 7)^5$ = negative number! This changes the value completely instead of making it a reciprocal. Always move the exponent to negative: $(10 \times 7)^{-5}$ means reciprocal.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

What's the difference between negative exponents and negative numbers?

A negative exponent like $x^{-5}$ means reciprocal (flip the fraction), while a negative number like $-x^5$ just means opposite sign!

Do I need to multiply out (10 × 7) first?

No! Keep the base as $(10 \times 7)$ since the question asks for the expression form. The parentheses show it's treated as one unit.

Why isn't the answer just 1/70^5?

While $10 \times 7 = 70$ , the question specifically asks to insert the corresponding expression, so we keep the format $(10 \times 7)^{-5}$ .

How do I remember the negative exponent rule?

Think: "Flip it, don't negate it!" When you see $\frac{1}{something}$ , the something gets a negative exponent to move upstairs.

Can I expand this to 10^(-5) × 7^(-5)?

Yes! Using the rule $(ab)^n = a^n b^n$ , we get $(10 \times 7)^{-5} = 10^{-5} \times 7^{-5}$ , but the given answer choices want the compact form.

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