Problem 33 - Algebra 1

Question

Write your answer as a power or as a product of powers. $$ \left[(5+x)^{3}\right]^{6} $$

Step-by-Step Solution

Answer

The simplified expression is \((5 + x)^{18}\)

Step 1: Identify the base and the exponents

The base in this problem is \((5+x)\) and the two exponents are 3 and 6.

Step 2: Apply the power of a power rule

The power of a power rule states that \((a^n)^m\) can be simplified as \(a^{n \times m}\). Applying this rule to our expression gives \((5 + x)^{3 \times 6}\).

Step 3: Multiply the exponents

Multiply 3 by 6 to obtain the final exponent. This gives \((5 + x)^{18}\).

Key Concepts

Power of a Power Rule

The power of a power rule is an essential concept in exponents. It helps simplify expressions where exponents are involved in more than one layer. When you see an expression like

• \( (a^n)^m \),

the power of a power rule allows us to break it down to a simpler form

• \( a^{n \times m} \).

This rule is incredibly useful because it prevents the need to expand and repeatedly multiply the base by itself. In our example,

• \([ (5 + x)^3 ]^6\),

by using the power of a power rule, we can directly multiply the two exponents, 3 and 6. This makes the expression much simpler, becoming

• \( (5 + x)^{18} \).

Base and Exponent Identification

Understanding the components of an exponential expression is crucial before simplifying it. Every exponential expression can be broken down into two key parts:

• The base: The number or expression that is being multiplied by itself.
• The exponent: The power to which the base is raised, indicating how many times the base will be multiplied by itself.

In our exercise, the base is clearly defined as

• \((5 + x)\),

and the exponents involved are

• 3 and 6.

Noticing these elements helps in applying the correct rules, such as the power of a power rule, to simplify the expression confidently and correctly.

Multiplying Exponents

Multiplying exponents becomes straightforward once you've understood how to handle a power raised to another power. This occurs after identifying the exponents in your expression. Knowing that multiplying the exponents is required, you simply take the two numbers and calculate their product. In our particular problem:

• The exponents are 3 and 6 from the expression \([ (5 + x)^3 ]^6\).

By multiplying these exponents, you perform the calculation

• \(3 \times 6 = 18\),

resulting in a single, simplified exponent. This results in the expression now written as

• \( (5 + x)^{18} \).

Multiplying exponents helps keep expressions neat and concise, revealing the true power of exponent rules in simplifying mathematics.