Simplify the Fraction: a³ divided by (3³ × 5³)

Q: Why are both expressions a' and b' considered correct?

Both expressions are mathematically equivalent! $ \frac{a^3}{(3 \times 5)^3} $ and $ \left(\frac{a}{3 \times 5}\right)^3 $ represent the same value using the rule $ \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n $.

Q: How do I know when to use the rule $ a^n \times b^n = (ab)^n $ ?

Use this rule when you see the same exponent on different bases being multiplied together. In our case, $ 3^3 \times 5^3 $ becomes $ (3 \times 5)^3 = 15^3 $.

Q: Can I simplify $ \frac{a^3}{3^3 \times 5^3} $ to $ \frac{a}{3 \times 5} $ ?

No! That would cancel the exponents incorrectly. The correct simplification keeps the cubed relationship: $ \frac{a^3}{15^3} = \left(\frac{a}{15}\right)^3 $, not just $ \frac{a}{15} $.

Q: What's the difference between $ (3 \times 5)^3 $ and $ 3^3 \times 5^3 $ ?

There's no difference - they're equal! $ (3 \times 5)^3 = 15^3 = 3375 $ and $ 3^3 \times 5^3 = 27 \times 125 = 3375 $. This is why both forms are acceptable.

Q: How do I verify my equivalent expressions are correct?

Substitute a specific value for the variable! Try $ a = 2 $: both $ \frac{8}{3375} $ and $ \left(\frac{2}{15}\right)^3 = \frac{8}{3375} $ give the same result.

Exponent Rules with Equivalent Expressions

Insert the corresponding expression:

$\frac{a^3}{3^3\times5^3}=$

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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, a product raised to the power (N)

00:09 equals the product broken down into factors where each factor is raised to power (N)

00:17 We will apply this formula to our exercise, in the opposite direction

00:21 We'll combine the factors into a multiplication operation inside of parentheses

00:25 According to the laws of exponents, a fraction raised to the power (N)

00:29 equals a fraction where both the numerator and denominator are in power (N)

00:34 We will apply this formula to our exercise, in the opposite direction

00:38 We'll place the entire fraction inside of parentheses and raise it to the appropriate power

00:42 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:

$\frac{a^3}{3^3\times5^3}=$

Step-by-step solution

To solve this problem, we need to simplify and express the equation $\frac{a^3}{3^3 \times 5^3}$ using exponent rules.

The denominator $3^3 \times 5^3$ can be simplified using the property of exponents: $(ab)^n = a^n \times b^n$ . This means that:

$3^3\times5^3=(3\times5)^3$ .

Therefore, the expression can be rewritten as:

$\frac{a^3}{(3\times5)^3 }$ which is actually the same as $\left(\frac{a}{3\times5}\right)^3$ , using the identity $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ .

Thus, the expression can also be written as:

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

Looking at the provided choices, this expression corresponds to choice marked as $\left(\frac{a}{3 \times 5}\right)^3$ .

Therefore, the expression matches both rewritten forms:

The correct answer is a'+b' are correct

Final Answer

a'+b' are correct

Key Points to Remember

Essential concepts to master this topic

Product Rule: $a^n \times b^n = (a \times b)^n$ applies in reverse too
Technique: $3^3 \times 5^3 = (3 \times 5)^3 = 15^3$ simplifies the denominator
Check: Verify $\frac{a^3}{b^3} = \left(\frac{a}{b}\right)^3$ gives same result ✓

Common Mistakes

Avoid these frequent errors

Treating multiple choice answers as mutually exclusive
Don't assume only one answer can be correct when dealing with equivalent expressions! $\frac{a^3}{(3 \times 5)^3}$ and $\left(\frac{a}{3 \times 5}\right)^3$ are mathematically identical. Always check if expressions are equivalent before choosing just one answer.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why are both expressions a' and b' considered correct?

Both expressions are mathematically equivalent! $\frac{a^3}{(3 \times 5)^3}$ and $\left(\frac{a}{3 \times 5}\right)^3$ represent the same value using the rule $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ .

How do I know when to use the rule $a^n \times b^n = (ab)^n$ ?

Use this rule when you see the same exponent on different bases being multiplied together. In our case, $3^3 \times 5^3$ becomes $(3 \times 5)^3 = 15^3$ .

Can I simplify $\frac{a^3}{3^3 \times 5^3}$ to $\frac{a}{3 \times 5}$ ?

No! That would cancel the exponents incorrectly. The correct simplification keeps the cubed relationship: $\frac{a^3}{15^3} = \left(\frac{a}{15}\right)^3$ , not just $\frac{a}{15}$ .

What's the difference between $(3 \times 5)^3$ and $3^3 \times 5^3$ ?

There's no difference - they're equal! $(3 \times 5)^3 = 15^3 = 3375$ and $3^3 \times 5^3 = 27 \times 125 = 3375$ . This is why both forms are acceptable.

How do I verify my equivalent expressions are correct?

Substitute a specific value for the variable! Try $a = 2$ : both $\frac{8}{3375}$ and $\left(\frac{2}{15}\right)^3 = \frac{8}{3375}$ give the same result.

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Simplify the Fraction: a³ divided by (3³ × 5³)

Exponent Rules with Equivalent Expressions

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why are both expressions a' and b' considered correct?

How do I know when to use the rule $a^n \times b^n = (ab)^n$ ?

Can I simplify $\frac{a^3}{3^3 \times 5^3}$ to $\frac{a}{3 \times 5}$ ?

What's the difference between $(3 \times 5)^3$ and $3^3 \times 5^3$ ?

How do I verify my equivalent expressions are correct?

🌟 Unlock Your Math Potential

More Questions

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Complete the Expression: (a⁵x⁵)/(7⁵b⁵) Algebraic Fraction

Simplify the Fraction: a³ divided by (3³ × 5³)

Exponent Rules with Equivalent Expressions

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why are both expressions a' and b' considered correct?

How do I know when to use the rule an×bn=(ab)n a^n \times b^n = (ab)^n an×bn=(ab)n?

Can I simplify a333×53 \frac{a^3}{3^3 \times 5^3} 33×53a3​ to a3×5 \frac{a}{3 \times 5} 3×5a​?

What's the difference between (3×5)3 (3 \times 5)^3 (3×5)3 and 33×53 3^3 \times 5^3 33×53?

How do I verify my equivalent expressions are correct?

🌟 Unlock Your Math Potential

More Questions

Powers of a Fraction

Simplify the Expression: 4^6 Divided by (a^6 × x^6)

Simplify the Expression: (7²x²)/a² Algebraic Fraction

Simplify the Quartic Fraction: x⁴/a⁴ Expression Problem

Complete the Expression: (a⁵x⁵)/(7⁵b⁵) Algebraic Fraction

How do I know when to use the rule $a^n \times b^n = (ab)^n$ ?

Can I simplify $\frac{a^3}{3^3 \times 5^3}$ to $\frac{a}{3 \times 5}$ ?

What's the difference between $(3 \times 5)^3$ and $3^3 \times 5^3$ ?