Simplify the Fraction: (4³)/(5³×7³) Step-by-Step Solution

Q: Why can I combine 5³ × 7³ into (5 × 7)³?

This uses the power of a product rule: $ a^n \times b^n = (a \times b)^n $. Since both 5 and 7 have the same exponent (3), you can factor out the common exponent!

Q: How do I know when to use the power of quotient rule?

Look for fractions where both numerator and denominator have the same exponent. When you see $ \frac{a^n}{b^n} $, you can always rewrite it as $ \left(\frac{a}{b}\right)^n $.

Q: What's the difference between the answer choices?

Choice A: $ \frac{4}{(5 \times 7)^3} $ - exponent only on denominatorChoice B: $ \left(\frac{4}{5 \times 7}\right)^3 $ - correct, exponent on entire fractionChoice C adds an extra 3 to the numerator, which is wrong!

Q: Can I solve this by calculating 4³, 5³, and 7³ separately?

Yes, but it's much harder! You'd get $ \frac{64}{125 \times 343} = \frac{64}{42875} $. The power of quotient rule gives the same result but in a much simpler form.

Q: Why do we want the simplified form instead of calculating everything?

Algebraic expressions are often more useful than decimal approximations. The form $ \left(\frac{4}{35}\right)^3 $ clearly shows the structure and is easier to work with in further calculations.

Power of Quotients with Product Denominators

Insert the corresponding expression:

$\frac{4^3}{5^3\times7^3}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's simplify this problem together.

00:12 Remember that when a product is raised to a power, N, it's like raising each factor to that same power.

00:19 So, we'll break it down and apply this to our exercise. Watch as we use parentheses.

00:24 Now, let's talk about raising a fraction to a power, N.

00:29 This means both the top and bottom numbers are raised to that power, N.

00:34 We'll use this idea to write our expression as a fraction with a power in parentheses.

00:41 And there you have it!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{4^3}{5^3\times7^3}=$

Step-by-step solution

To solve this problem, we will apply the rule for powers of a quotient. By recognizing that the expression $\frac{4^3}{5^3 \times 7^3}$ can be seen in terms of powers, we can reformulate it:

$\frac{4^3}{5^3 \times 7^3} = \frac{4^3}{(5 \times 7)^3}$

Now, we see this as a single fraction raised to the same power, which can be expressed using the power of a fraction rule:

$\frac{4^3}{(5 \times 7)^3} = \left(\frac{4}{5 \times 7}\right)^3$

Thus, the expression given is equivalent to

$\left(\frac{4}{5 \times 7}\right)^3$ .

Final Answer

$\left(\frac{4}{5\times7}\right)^3$

Key Points to Remember

Essential concepts to master this topic

Rule: Use power of quotient property: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
Technique: Recognize $5^3 \times 7^3 = (5 \times 7)^3$ using power of product rule
Check: Verify $\left(\frac{4}{35}\right)^3 = \frac{64}{42875}$ equals original expression ✓

Common Mistakes

Avoid these frequent errors

Distributing the exponent only to numerator
Don't write $\frac{4^3}{5^3 \times 7^3}$ as $\frac{4}{5 \times 7}^3$ = wrong placement of exponent! This makes the exponent apply only to 7, not the entire denominator. Always ensure the exponent applies to the complete fraction by using parentheses: $\left(\frac{4}{5 \times 7}\right)^3$ .

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why can I combine 5³ × 7³ into (5 × 7)³?

This uses the power of a product rule: $a^n \times b^n = (a \times b)^n$ . Since both 5 and 7 have the same exponent (3), you can factor out the common exponent!

How do I know when to use the power of quotient rule?

Look for fractions where both numerator and denominator have the same exponent. When you see $\frac{a^n}{b^n}$ , you can always rewrite it as $\left(\frac{a}{b}\right)^n$ .

What's the difference between the answer choices?

Choice A: $\frac{4}{(5 \times 7)^3}$ - exponent only on denominator
Choice B: $\left(\frac{4}{5 \times 7}\right)^3$ - correct, exponent on entire fraction
Choice C adds an extra 3 to the numerator, which is wrong!

Can I solve this by calculating 4³, 5³, and 7³ separately?

Yes, but it's much harder! You'd get $\frac{64}{125 \times 343} = \frac{64}{42875}$ . The power of quotient rule gives the same result but in a much simpler form.

Why do we want the simplified form instead of calculating everything?

Algebraic expressions are often more useful than decimal approximations. The form $\left(\frac{4}{35}\right)^3$ clearly shows the structure and is easier to work with in further calculations.

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