Evaluate (2/105)^(a+2): Complex Fraction with Variable Exponent

Q: Why does the exponent apply to the whole denominator?

Because of the parentheses grouping! When you have $ (3\times7\times5) $ in the denominator, the exponent applies to this entire product, not just individual factors.

Q: What's the difference between the wrong and right answers?

The wrong answer $ \frac{2^{a+2}}{3\times7\times5^{a+2}} $ only raises 5 to the power, while the correct answer $ \frac{2^{a+2}}{(3\times7\times5)^{a+2}} $ raises the entire denominator to the power.

Q: How do I remember when to use parentheses with exponents?

Think of it this way: exponents are like multiplication - they distribute over everything inside parentheses. If there's no parentheses separating factors, the exponent only applies to what's directly attached.

Q: Can I simplify 3×7×5 first?

You could calculate $ 3\times7\times5 = 105 $ to get $ \frac{2^{a+2}}{105^{a+2}} $, but the question asks for the expression as given, so keep it in factored form.

Q: What if I forget the exponent rule?

Remember: "Power of a fraction = fraction of powers". The exponent on the outside affects both the top and bottom separately, like breaking apart a sandwich!

Exponent Rules with Fraction Bases

Insert the corresponding expression:

$\left(\frac{2}{3\times7\times5}\right)^{a+2}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 According to the laws of exponents, a fraction raised to the power (N)

00:07 equals the numerator and denominator raised to the same power (N)

00:10 We will apply this formula to our exercise

00:13 Note that the entire exponent contains an addition operation

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:

$\left(\frac{2}{3\times7\times5}\right)^{a+2}=$

Step-by-step solution

To solve the problem, we will leverage the rules of exponents:

Apply the rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ to the expression $\left(\frac{2}{3\times7\times5}\right)^{a+2}$ .
Distribute the exponent $(a+2)$ to both the numerator and the denominator.

First, distribute the exponent to the numerator:

$2^{a+2}$

Now, distribute the exponent to the entire denominator:

$(3\times7\times5)^{a+2}$

Thus, the expression becomes:

$\frac{2^{a+2}}{(3\times7\times5)^{a+2}}$

This matches choice 3: $\frac{2^{a+2}}{\left(3\times7\times5\right)^{a+2}}$ .

Final Answer

$\frac{2^{a+2}}{\left(3\times7\times5\right)^{a+2}}$

Key Points to Remember

Essential concepts to master this topic

Rule: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ applies exponent to both parts
Technique: Distribute $(a+2)$ to get $\frac{2^{a+2}}{(3\times7\times5)^{a+2}}$
Check: Verify denominator is $(3\times7\times5)^{a+2}$ not $3\times7\times5^{a+2}$ ✓

Common Mistakes

Avoid these frequent errors

Only applying exponent to one factor in denominator
Don't apply the exponent only to the last factor like $3\times7\times5^{a+2}$ = wrong grouping! This ignores the parentheses that group all factors together. Always apply the exponent to the entire expression in parentheses: $(3\times7\times5)^{a+2}$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does the exponent apply to the whole denominator?

Because of the parentheses grouping! When you have $(3\times7\times5)$ in the denominator, the exponent applies to this entire product, not just individual factors.

What's the difference between the wrong and right answers?

The wrong answer $\frac{2^{a+2}}{3\times7\times5^{a+2}}$ only raises 5 to the power, while the correct answer $\frac{2^{a+2}}{(3\times7\times5)^{a+2}}$ raises the entire denominator to the power.

How do I remember when to use parentheses with exponents?

Think of it this way: exponents are like multiplication - they distribute over everything inside parentheses. If there's no parentheses separating factors, the exponent only applies to what's directly attached.

Can I simplify 3×7×5 first?

You could calculate $3\times7\times5 = 105$ to get $\frac{2^{a+2}}{105^{a+2}}$ , but the question asks for the expression as given, so keep it in factored form.

What if I forget the exponent rule?

Remember: "Power of a fraction = fraction of powers". The exponent on the outside affects both the top and bottom separately, like breaking apart a sandwich!

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