Evaluate the Expression: (5×3)^6x with Exponential Rules

Q: Why does the exponent apply to both numbers?

The power of product rule states that when you raise a product to a power, each factor gets that power. Think of it like this: $ (5 \times 3)^{6x} $ means multiply (5×3) by itself 6x times, which is the same as multiplying 5 by itself 6x times AND 3 by itself 6x times!

Q: What if there are more than two numbers in parentheses?

The same rule applies! For example, $ (2 \times 4 \times 7)^3 = 2^3 \times 4^3 \times 7^3 $. Every single base inside the parentheses gets the exponent.

Q: Does this work when the exponent has variables like 6x?

Absolutely! Variable exponents follow the same rules as number exponents. The expression 6x is treated as one complete exponent that gets distributed to each base.

Q: How is this different from (5+3)^6x?

Be careful! The power of product rule only works with multiplication. For addition like $ (5+3)^{6x} $, you must first simplify inside parentheses: $ (8)^{6x} = 8^{6x} $. You cannot distribute exponents over addition!

Q: Can I simplify 5×3 first before applying the exponent?

You could write $ (15)^{6x} = 15^{6x} $, but the question asks you to show the distributed form. Both $ 15^{6x} $ and $ 5^{6x} \times 3^{6x} $ are mathematically equal, but follow the format requested in the problem.

Power of Product with Variable Exponents

Insert the corresponding expression:

$\left(5\times3\right)^{6x}=$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 In order to open parentheses with a multiplication operation and an outside exponent

00:07 Raise each factor to the power

00:10 We'll apply this formula to our exercise

00:13 Note that all the factors inside of the parentheses are raised to the same power(N)

00:17 Therefore we'll raise each factor to this power (N)

00:21 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:

$\left(5\times3\right)^{6x}=$

Step-by-step solution

To solve this problem, we need to apply the power of a product rule to the given expression, $(5 \times 3)^{6x}$ .

According to the exponent rule for the power of a product, when an expression in the form $(a \times b)^n$ is encountered, it can be expanded to $a^n \times b^n$ . Here, $a = 5$ , $b = 3$ , and $n = 6x$ .

By applying this rule, we distribute the exponent $6x$ to each of the bases within the parentheses:

First, apply the exponent to 5, resulting in $5^{6x}$ .
Then, apply the exponent to 3, resulting in $3^{6x}$ .

Thus, the expression $(5 \times 3)^{6x}$ simplifies to $5^{6x} \times 3^{6x}$ . This shows that the exponent $6x$ is correctly applied to each element within the parentheses.

Therefore, the expression $(5 \times 3)^{6x}$ is equivalent to $5^{6x} \times 3^{6x}$ .

Final Answer

$5^{6x}\times3^{6x}$

Key Points to Remember

Essential concepts to master this topic

Rule: Distribute exponent to each base: $(a \times b)^n = a^n \times b^n$
Technique: Apply 6x to both 5 and 3: $(5 \times 3)^{6x} = 5^{6x} \times 3^{6x}$
Check: Each base gets the full exponent: both 5 and 3 raised to 6x ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only one base
Don't write $5^{6x} \times 3$ or $5 \times 3^{6x}$ = incomplete distribution! This ignores the power of product rule and gives wrong results. Always distribute the exponent to every single base inside the parentheses.

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 both numbers?

The power of product rule states that when you raise a product to a power, each factor gets that power. Think of it like this: $(5 \times 3)^{6x}$ means multiply (5×3) by itself 6x times, which is the same as multiplying 5 by itself 6x times AND 3 by itself 6x times!

What if there are more than two numbers in parentheses?

The same rule applies! For example, $(2 \times 4 \times 7)^3 = 2^3 \times 4^3 \times 7^3$ . Every single base inside the parentheses gets the exponent.

Does this work when the exponent has variables like 6x?

Absolutely! Variable exponents follow the same rules as number exponents. The expression 6x is treated as one complete exponent that gets distributed to each base.

How is this different from (5+3)^6x?

Be careful! The power of product rule only works with multiplication. For addition like $(5+3)^{6x}$ , you must first simplify inside parentheses: $(8)^{6x} = 8^{6x}$ . You cannot distribute exponents over addition!

Can I simplify 5×3 first before applying the exponent?

You could write $(15)^{6x} = 15^{6x}$ , but the question asks you to show the distributed form. Both $15^{6x}$ and $5^{6x} \times 3^{6x}$ are mathematically equal, but follow the format requested in the problem.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Exponents Rules questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime