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Chapter 6: Problem 126

Simplify each expression. (a) \((3 x)^{2}(5 x)\) (b) \(\left(5 t^{2}\right)^{3}(3 t)^{2}\)

Short Answer

Expert verified
(a) \( 45x^3 \), (b) \( 1125t^8 \)

Step by step solution

01

Simplify \( (3x)^2 \)

First, apply the power rule to \( (3x)^2 \). This means you will square both the coefficient and the variable. So, \( (3x)^2 = (3^2)(x^2) = 9x^2 \).
02

Multiply \( 9x^2 \) by \( 5x \)

Next, multiply the result from Step 1 by \( 5x \). This requires multiplying the coefficients and then adding the exponents of the same base: \( 9x^2 \times 5x = 9 \times 5 \times x^{2+1} = 45x^3 \).
03

Simplify \( \left(5t^2\right)^3 \)

Now for the second expression, apply the power rule to \( \left(5t^2\right)^3 \). You raise both the coefficient and the variable to the power of 3: \( \left(5t^2\right)^3 = 5^3 (t^2)^3 = 125t^6 \).
04

Simplify \( (3t)^2 \)

Similarly, apply the power rule to \( (3t)^2 \), resulting in \( (3t)^2 = 3^2 t^2 = 9t^2 \).
05

Multiply \( 125t^6 \) by \( 9t^2 \)

Finally, multiply the result from Step 3 by the result from Step 4: \( 125t^6 \times 9t^2 = 125 \times 9 \times t^{6+2} = 1125t^8 \).

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Key Concepts

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

power rule
When simplifying algebraic expressions, the power rule is a key concept. The power rule states that when raising a power to another power, you multiply the exponents. For example, when you have \( (a^m)^n \), the simplified form is \( a^{m \times n} \).
Here's how it applies:
  • In Step 1 of our exercise, \( (3x)^2 \), we square both the coefficient and the variable. This gives us \( (3^2)(x^2) = 9x^2 \).
  • Similarly, in Step 3, \( \left(5t^2\right)^3 \), we raise the coefficient and the variable term to the third power: \( (5^3) (t^2)^3 = 125t^6 \).
Understanding the power rule makes it easier to simplify expressions correctly.
multiplying coefficients
An important aspect of algebraic simplification is multiplying coefficients. Coefficients are the numerical values in front of variables. Here鈥檚 what to keep in mind:
  • To multiply coefficients, you simply multiply the numbers together.
  • For example, in Step 2 of our exercise, we multiply the coefficients 9 and 5: \( 9 \times 5 = 45 \).
  • Again, in Step 5, we multiply the coefficients 125 and 9, resulting in \( 125 \times 9 = 1125 \).
This straightforward multiplication is critical for accurate simplification of expressions.
Mastering this will also help when coefficients are larger or more complex.
adding exponents
When multiplying variables with the same base, you add the exponents. This is essential in simplifying expressions. Here's a closer look:
  • Suppose you have \( x^m \times x^n \), you'll combine them as \ x^{m+n} \.
  • In Step 2, we multiply \ x^2 \ by \ x \; because \ x \ alone is \ x^1 \, it becomes \ x^{2+1} = x^3 \.
  • In Step 5, multiplying \ t^6 \ and \ t^2 \, it results in \ t^{6+2} = t^8 \.
Adding exponents is essential for properly combining terms and obtaining the correct expression. Practice with various examples will strengthen your understanding of this concept.

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Most popular questions from this chapter

Divide each polynomial by the monomial. $$\frac{49 c^{2} d^{2}-70 c^{3} d^{3}-35 c^{2} d^{4}}{7 c d^{2}}$$

Simplify. (a) \(3^{-4}\) (b) \(10^{-2}\)

Divide each polynomial by the binomial. $$\left(p^{2}+11 p+16\right) \div(p+8)$$

Simplify. (a) \(3 \cdot 4^{-2}\) (b) \((3 \cdot 4)^{-2}\)

Mixed Practice $$\frac{56 y^{4} z^{5}}{7 y^{3} z^{3}}-\frac{45 y^{2} z^{2}}{5 y}$$
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