Expand (2a)^(y+5): Step-by-Step Expression Expansion

Q: What's the difference between (2a)^5 and 2a^5?

$ (2a)^5 = 2^5 \times a^5 = 32a^5 $, but $ 2a^5 $ just means 2 times a^5. The parentheses make a huge difference - they group 2 and a together as one base!

Q: Do I need to simplify 2^(y+5) further?

No! $ 2^{y+5} $ is already in its simplest form when the exponent contains variables. You cannot break it down further without knowing specific values for y.

Q: Can this rule work with more than two factors?

Absolutely! For example: $ (3xy)^4 = 3^4 \times x^4 \times y^4 $. The power rule applies to any number of factors in the base.

Q: What if the base has negative numbers?

The rule still works! $ (-2a)^{y+5} = (-2)^{y+5} \times a^{y+5} $. Just remember that negative bases can give positive or negative results depending on whether the exponent is even or odd.

Power of Products with Variable Exponents

Expand the following equation:

$\left(2a\right)^{y+5}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

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

00:07 equals the product where each factor is raised to the same power (N)

00:10 We will apply this formula to our exercise

00:14 Break down the product into factors and raise to the power (N)

00:17 Note that the power (N) contains an addition operation

00:21 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Expand the following equation:

$\left(2a\right)^{y+5}=$

Step-by-step solution

To solve the problem of expanding $(2a)^{y+5}$ , we'll use the Power of a Product Rule.

Step 1: Identify the base and the exponent. Here, the base is $2a$ , and the exponent is $y+5$ .
Step 2: Apply the Power of a Product Rule, which states that $(ab)^n = a^n \times b^n$ . In this case, apply it to the base $2a$ .
Step 3: Expand the expression: $(2a)^{y+5} = 2^{y+5} \times a^{y+5}$ .

By applying the rule, we separate the exponential expression into two parts, one for each component of the base:

$(2a)^{y+5} = 2^{y+5} \times a^{y+5}$

This result shows that both $2$ and $a$ are individually raised to the power of $y+5$ . The application of the product rule ensures that each base component is treated equally within the exponentiation.

Therefore, the expanded form of the expression is $2^{y+5} \times a^{y+5}$ , which corresponds to answer choice 4.

Final Answer

$2^{y+5}\times a^{y+5}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Apply $(ab)^n = a^n \times b^n$ to separate base components
Technique: Distribute exponent to each factor: $(2a)^{y+5} = 2^{y+5} \times a^{y+5}$
Check: Both 2 and a must have identical exponent (y+5) ✓

Common Mistakes

Avoid these frequent errors

Only applying exponent to one factor
Don't write $(2a)^{y+5} = 2a^{y+5}$ = wrong expansion! This applies the exponent only to 'a' while leaving '2' unchanged, violating the power rule. Always apply the exponent to every factor in the base using $(ab)^n = a^n \times b^n$ .

Practice Quiz

Test your knowledge with interactive questions

$$ (4^2)^3+(g^3)^4= $$

FAQ

Everything you need to know about this question

Why can't I just write 2a^(y+5)?

Because that only applies the exponent to a, not to 2! The expression $(2a)^{y+5}$ means the entire product 2a is raised to the power y+5, so both factors need the exponent.

What's the difference between (2a)^5 and 2a^5?

$(2a)^5 = 2^5 \times a^5 = 32a^5$ , but $2a^5$ just means 2 times a^5. The parentheses make a huge difference - they group 2 and a together as one base!

Do I need to simplify 2^(y+5) further?

No! $2^{y+5}$ is already in its simplest form when the exponent contains variables. You cannot break it down further without knowing specific values for y.

Can this rule work with more than two factors?

Absolutely! For example: $(3xy)^4 = 3^4 \times x^4 \times y^4$ . The power rule applies to any number of factors in the base.

What if the base has negative numbers?

The rule still works! $(-2a)^{y+5} = (-2)^{y+5} \times a^{y+5}$ . Just remember that negative bases can give positive or negative results depending on whether the exponent is even or odd.

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