Calculate (2×5×6)^15: Evaluating a Product Raised to a Power

Q: What if one of the numbers was already raised to a power?

Use the power of a power rule! If you had $ (2^3\times5)^{15} $, it becomes $ 2^{45}\times5^{15} $ because 3 × 15 = 45.

Q: Do I need to calculate the actual numerical value?

Not for this type of question! The goal is to apply the exponent rule correctly and show the expanded form. Calculating $ 2^{15} $ would give huge numbers.

Q: What's the difference between this and distributing multiplication?

Great question! Distributing applies to addition/subtraction like $ 3(x + 2) $. Here we're using the power of products rule for exponents with multiplication.

Q: How do I remember this rule?

Think: "Power goes to everyone in the group!" Just like if a teacher gives homework to the whole class, the exponent gets applied to each factor in the parentheses.

Exponent Laws with Product Rule Application

Choose the expression that corresponds to the following:

$\left(2\times5\times6\right)^{15}=$

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

00:08 Each factor can be raised to the power (N)

00:14 We will 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:

$\left(2\times5\times6\right)^{15}=$

Step-by-step solution

We need to apply the power of products exponent rule. According to this rule, when we raise a product to a power, we can raise each individual term in the product to that power.

Mathematically, this is expressed as:

$(a \times b \times c)^n = a^n \times b^n \times c^n$

We can apply the power of a product rule to our expression:

Raise each factor inside the parentheses to the power of 15.

This gives us:

$2^{15} \times 5^{15} \times 6^{15}$

Therefore, the expression is $2^{15} \times 5^{15} \times 6^{15}$ .

Final Answer

$2^{15}\times5^{15}\times6^{15}$

Key Points to Remember

Essential concepts to master this topic

Power of Products Rule: Raise each factor to the given power separately
Technique: $(2\times5\times6)^{15} = 2^{15}\times5^{15}\times6^{15}$
Check: Each base number appears exactly once with exponent 15 ✓

Common Mistakes

Avoid these frequent errors

Only raising some factors to the power
Don't write $2^{15}\times5\times6^{15}$ = leaves middle factor unchanged! This violates the power of products rule and gives an incorrect expression. Always raise every single factor inside the parentheses to the outside power.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I just multiply 2×5×6 first, then raise to the 15th power?

You absolutely can! Both methods give the same result. The question asks for the expanded form using the power of products rule: $2^{15}\times5^{15}\times6^{15}$ .

What if one of the numbers was already raised to a power?

Use the power of a power rule! If you had $(2^3\times5)^{15}$ , it becomes $2^{45}\times5^{15}$ because 3 × 15 = 45.

Do I need to calculate the actual numerical value?

Not for this type of question! The goal is to apply the exponent rule correctly and show the expanded form. Calculating $2^{15}$ would give huge numbers.

What's the difference between this and distributing multiplication?

Great question! Distributing applies to addition/subtraction like $3(x + 2)$ . Here we're using the power of products rule for exponents with multiplication.

How do I remember this rule?

Think: "Power goes to everyone in the group!" Just like if a teacher gives homework to the whole class, the exponent gets applied to each factor in the parentheses.

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