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Chapter 0: Problem 5

Simplify the expression. \(\left(-3 x^{5} y^{2}\right)\left(2 x^{-3} y\right)\)

Short Answer

Expert verified
-6x^{2}y^{3}

Step by step solution

01

- Multiply the Coefficients

First, multiply the numerical coefficients of the two terms. Here we have \(-3 \times 2 = -6\).
02

- Apply the Power Rule for Exponents on x

Combine the exponents of the term with the base \(x\). Using the power rule \(a^m \times a^n = a^{m+n}\), calculate \(x^{5} \times x^{-3} = x^{5+(-3)} = x^{2}\).
03

- Apply the Power Rule for Exponents on y

Similarly, combine the exponents of the term with the base \(y\). Calculate \(y^2 \times y = y^{2+1} = y^3\).
04

- Write the Final Expression

Combining the results from steps 1 to 3, the simplified expression is \(-6x^{2}y^{3}\).

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Key Concepts

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

Multiplying Coefficients
To simplify algebraic expressions, the first step is often to handle the numerical coefficients. Coefficients are the numbers in front of the variables. In our example, we have \(-3\) and \(2\). These should be multiplied together directly. Remember that multiplying two negative numbers results in a positive number, but here we have one negative and one positive: \(-3 \times 2 = -6\). This gives us the coefficient for our final simplified expression.
Power Rule for Exponents
The power rule for exponents is essential in simplifying expressions involving exponents. It states that when you multiply like bases, you add the exponents: \(a^m \times a^n = a^{m+n}\).
In our example, we have bases of \(x\) and \(y\). Let's start with \(x\). We combine the exponents of \(x^5\) and \(x^{-3}\). So, using the power rule: \(x^5 \times x^{-3} = x^{5+(-3)} = x^2\).
Next, we combine the exponents of \(y^2\) and \(y\). This is simpler since \(y\) can be written as \(y^1\) (every variable has an invisible exponent of 1 if no exponent is shown). Thus: \(y^2 \times y^1 = y^{2+1} = y^3\). This handles our exponents correctly.
Combining Like Terms
Combining like terms is the final step in simplifying the expression. Like terms are terms that have the same variables raised to the same power. In our case, after handling the coefficients and exponents, we combine everything:
  • The coefficient part: \(-6\)
  • The \(x\) part: \(x^2\)
  • The \(y\) part: \(y^3\)
Putting it all together, we get the simplified expression: \-6x^2 y^3.
This process of combining like terms ensures that we end up with the simplest form of the algebraic expression.

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Most popular questions from this chapter

A polynomial in the variable \(x\) is defined as an expression of the form \(a_{n} x^{n}+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+\cdots+a_{1} x+a_{0} .\) Explain what this means.

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (\sqrt{5+2 \sqrt{x}})(\sqrt{5-2 \sqrt{x}}) $$

Use a calculator to approximate the expression to 2 decimal places. $$\frac{6-3 \sqrt{5}}{4}$$

Simplify each expression. $$ (y+7)^{2}-2(y-3)^{2} $$

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (2 v-\sqrt{17})^{2} $$
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