Simplify the Product: 3^(5x+1/2) × 7^(5x+1/2) × 5^(5x+1/2)

Q: Why can I combine the bases but keep the same exponent?

This follows the Power of a Product Rule: $ a^m \times b^m = (a \times b)^m $. Since all three terms have the identical exponent $ 5x+\frac{1}{2} $, you can factor it out and combine the bases.

Q: What if the exponents were different?

If exponents differed, you cannot use this rule! For example, $ 3^2 \times 7^3 $ stays as is because the powers don't match. This rule only works when all exponents are identical.

Q: Do I need to calculate 3 × 7 × 5 = 105?

Not necessarily! The answer $ (3 \times 7 \times 5)^{5x+\frac{1}{2}} $ is perfectly acceptable. You could simplify to $ 105^{5x+\frac{1}{2}} $, but leaving it factored often makes the structure clearer.

Q: How do I know this rule applies here?

Look for the same exponent on different bases. Here, $ 5x+\frac{1}{2} $ appears on all three terms (3, 7, and 5), so you can use the power of a product rule.

Q: What's the difference between this and the product rule for same bases?

That's $ a^m \times a^n = a^{m+n} $ (add exponents, same base). Here we have $ a^m \times b^m = (ab)^m $ (multiply bases, same exponent). Different rules for different situations!

Exponent Rules with Same Powers

Reduce the following equation:

$3^{5x+\frac{1}{2}}\times7^{5x+\frac{1}{2}}\times5^{5x+\frac{1}{2}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to the laws of exponents, a product raised to the power (N)

00:10 Equals a product where each factor is raised to that same power (N)

00:14 We will apply this formula to our exercise

00:18 Note that the exponent (N) contains an addition operation

00:26 This is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Reduce the following equation:

$3^{5x+\frac{1}{2}}\times7^{5x+\frac{1}{2}}\times5^{5x+\frac{1}{2}}=$

Step-by-step solution

To solve this problem, we need to simplify the given product of exponentials. Let's follow the steps:

Step 1: Identify that each factor $3^{5x+\frac{1}{2}}$ , $7^{5x+\frac{1}{2}}$ , and $5^{5x+\frac{1}{2}}$ shares the same exponent $5x + \frac{1}{2}$ .
Step 2: Apply the Power of a Product Rule, which allows us to combine the product of exponents with the same power. The rule states: $a^{m} \times b^{m} \times c^{m} = (a \times b \times c)^{m}$ for $a = 3$ , $b = 7$ , and $c = 5$ .
Step 3: Combine the bases under a single exponent: $3^{5x+\frac{1}{2}} \times 7^{5x+\frac{1}{2}} \times 5^{5x+\frac{1}{2}} = (3 \times 7 \times 5)^{5x+\frac{1}{2}}$

Therefore, the reduced form of the given expression is $\left(3 \times 7 \times 5\right)^{5x+\frac{1}{2}}$ .

Final Answer

$\left(3\times7\times5\right)^{5x+\frac{1}{2}}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When bases differ but exponents match, combine bases first
Technique: $a^m \times b^m = (a \times b)^m$ gives $(3 \times 7 \times 5)^{5x+\frac{1}{2}}$
Check: Verify bases multiply to 105 and exponent stays $5x+\frac{1}{2}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of combining bases
Don't add the exponents to get $3 \times 7 \times 5^{15x+\frac{3}{2}}$ ! This confuses the product rule with the power rule and creates a completely different expression. Always combine the bases when exponents are identical: $(3 \times 7 \times 5)^{5x+\frac{1}{2}}$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I combine the bases but keep the same exponent?

This follows the Power of a Product Rule: $a^m \times b^m = (a \times b)^m$ . Since all three terms have the identical exponent $5x+\frac{1}{2}$ , you can factor it out and combine the bases.

What if the exponents were different?

If exponents differed, you cannot use this rule! For example, $3^2 \times 7^3$ stays as is because the powers don't match. This rule only works when all exponents are identical.

Do I need to calculate 3 × 7 × 5 = 105?

Not necessarily! The answer $(3 \times 7 \times 5)^{5x+\frac{1}{2}}$ is perfectly acceptable. You could simplify to $105^{5x+\frac{1}{2}}$ , but leaving it factored often makes the structure clearer.

How do I know this rule applies here?

Look for the same exponent on different bases. Here, $5x+\frac{1}{2}$ appears on all three terms (3, 7, and 5), so you can use the power of a product rule.

What's the difference between this and the product rule for same bases?

That's $a^m \times a^n = a^{m+n}$ (add exponents, same base). Here we have $a^m \times b^m = (ab)^m$ (multiply bases, same exponent). Different rules for different situations!

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