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

$112^0=\text{?}$

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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