Calculate the Expression: 5×3⁵×11⁵ Using Powers and Multiplication

Q: Why can't I include the 5 in the parentheses with the powers?

Because 5 doesn't have an exponent of 5! The rule $ a^n \times b^n = (a \times b)^n $ only works when both terms have the same exponent. Here, only 3 and 11 have the exponent 5.

Q: How do I know which terms can be factored together?

Look for terms with identical exponents. In $ 5\times3^5\times11^5 $, only $ 3^5 $ and $ 11^5 $ have the same exponent (5), so they can be combined.

Q: What's the difference between the power of a product and product of powers?

Power of a product: $ (ab)^n = a^n b^n $ (expanding)Product of powers: $ a^n b^n = (ab)^n $ (factoring)We used the second rule to factor the expression.

Q: Can I simplify further after getting the factored form?

Yes! You can calculate $ 3 \times 11 = 33 $, so the final simplified form would be $ 5 \times 33^5 $. Both forms are correct answers.

Q: What if the exponents were different numbers?

If the exponents don't match (like $ 3^4 \times 11^5 $), you cannot factor them together using this rule. The expression would stay as is.

Power Rules with Product Factoring

Choose the expression that corresponds to the following:

$5\times3^5\times11^5=$

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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 When we are presented with a multiplication operation where all the factors have the same exponent (N)

00:08 We can write the power (N) over the entire multiplication

00:12 We can apply this formula to our exercise

00:21 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the expression that corresponds to the following:

$5\times3^5\times11^5=$

Step-by-step solution

To solve the expression $5\times3^5\times11^5$ , we can apply the rule of exponents known as the power of a product rule. This rule states that for any integers $a$ , $b$ , and $n$ , $(a\times b)^n = a^n \times b^n$ .

Step 1: Analyse the expression
The expression we have is $5\times3^5\times11^5$ .

Step 2: Apply the Power of a Product rule
Notice that both 3 and 11 are raised to the power of 5. We can use the inverse of the power of a product formula to combine these terms:

$3^5 \times 11^5$ can be written as $(3 \times 11)^5$

Step 3: Rewrite the expression
Therefore, the expression $5\times3^5\times11^5$ becomes $5\times(3\times11)^5$ .

By applying the power of a product rule, we have determined that the equivalent expression for the given problem is $5\times(3\times11)^5$ .

Final Answer

$5\times\left(3\times11\right)^5$

Key Points to Remember

Essential concepts to master this topic

Product Power Rule: Same exponents can be factored together
Technique: Combine $3^5 \times 11^5 = (3 \times 11)^5$
Check: Verify by expanding back to original form ✓

Common Mistakes

Avoid these frequent errors

Factoring the coefficient with the powers
Don't write $5\times3^5\times11^5$ as $(5\times3\times11)^5$ = wrong factoring! The 5 has no exponent, so it stays separate from the powers. Always keep coefficients without exponents outside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that corresponds to the following:

$$ $\left(2\times11\right)^5=$

FAQ

Everything you need to know about this question

Why can't I include the 5 in the parentheses with the powers?

Because 5 doesn't have an exponent of 5! The rule $a^n \times b^n = (a \times b)^n$ only works when both terms have the same exponent. Here, only 3 and 11 have the exponent 5.

How do I know which terms can be factored together?

Look for terms with identical exponents. In $5\times3^5\times11^5$ , only $3^5$ and $11^5$ have the same exponent (5), so they can be combined.

What's the difference between the power of a product and product of powers?

Power of a product: $(ab)^n = a^n b^n$ (expanding)
Product of powers: $a^n b^n = (ab)^n$ (factoring)
We used the second rule to factor the expression.

Can I simplify further after getting the factored form?

Yes! You can calculate $3 \times 11 = 33$ , so the final simplified form would be $5 \times 33^5$ . Both forms are correct answers.

What if the exponents were different numbers?

If the exponents don't match (like $3^4 \times 11^5$ ), you cannot factor them together using this rule. The expression would stay as is.

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