Problem 158 Simplify each expression. $$\l... [FREE SOLUTION]

Chapter 6: Problem 158

Simplify each expression. $$\left(5 r^{2}\right)^{3}(3 r)^{2}$$

Short Answer

Expert verified

The simplified expression is $ 1125 r^{8} $.

Step by step solution

Apply the Power Rule

Use the power rule $(a^m)^n = a^{mn}$\ to simplify each term separately. For $ (5r^2)^3$: $ (5r^2)^3 = 5^3 \cdot (r^2)^3 = 125r^6$. For $ (3r)^2$: $ (3r)^2 = 3^2 \cdot r^2 = 9r^2$.

Multiply Simplified Terms Together

Multiply the simplified terms $125r^6$\

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule in Algebra

The power rule is an essential concept in algebra. It states that when you raise a power to another power, you multiply the exponents. Mathematically, this is expressed as $(a^m)^n = a^{mn}$.

In our example: $\left(5 r^{2}\right)^3$, we apply the power rule to each part separately:
- $ (5)^3 = 5 \times 5 \times 5 = 125 $
- $ (r^2)^3 = r^{2 \times 3} = r^6 $
By using the power rule, we simplify $(5r^2)^3$ to $125r^6$. Similarly, for $(3r)^2$:
- $ (3)^2 = 3 \times 3 = 9 $
- $ (r)^2 = r^2 $
This simplifies to $9r^2$.

Multiplication of Exponents

When multiplying expressions with exponents, if the bases are the same, you add the exponents. This rule is used often in algebra to simplify expressions. The general formula is \a^m \times a^n = a^{m+n}\.

In the given problem, we have already simplified the expressions to $125r^6$ and $9r^2$. Now we multiply them:
- Numbers: $125 \times 9 = 1125$
- Exponents: $r^6 \times r^2 = r^{6+2} = r^8$
So the final multiplication gives us: $125r^6 \times 9r^2 = 1125r^8$.

Simplifying Polynomial Expressions

Simplifying polynomial expressions involves combining like terms and applying algebraic rules to make the expression simpler.

In our example, we already simplified each part separately using the power rule, then multiplied using the exponent rules:
- $ (5r^2)^3 = 125r^6$
- $ (3r)^2 = 9r^2$
Then, combining them through multiplication:
- $ 125r^6 \times 9r^2 = 1125r^8$

The simplified polynomial expression is $1125r^8$. This means that by systematically applying algebraic rules, we can break the problem into manageable steps and simplify even complex expressions efficiently.

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91影视

Simplify each expression. $$\left(5 r^{2}\right)^{3}(3 r)^{2}$$

Short Answer

Step by step solution

Apply the Power Rule

Multiply Simplified Terms Together

Key Concepts

Power Rule in Algebra

Multiplication of Exponents

Simplifying Polynomial Expressions

One App. One Place for Learning.

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