Problem 33 Simplify. $$ -8 a_{2} b \cdo... [FREE SOLUTION]

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Elementary Algebra

John Redden

$Math Studyset 91影视 Explanations$ Math

2011 Edition

Chapter 5: Problem 33

Simplify. $$ -8 a_{2} b \cdot 2 a b $$

Short Answer

Expert verified

The simplified expression is $-16 a^3 b^2$.

Step by step solution

Identify the Terms and Multiply Coefficients

The expression given is $-8 a_{2} b \cdot 2 a b$. Firstly, identify the coefficients of each term: $-8$ and $2$. Multiply these coefficients together: $-8 \times 2 = -16$. This gives us the new coefficient for the simplified expression.

Apply the Laws of Exponents to the Variables

Now focus on the variables in the expression $a_2 b \cdot a b$. The laws of exponents tell us that when we multiply terms with the same base, we add their exponents. $a_{2} \cdot a = a^{2+1} = a^3$. Similarly, $b \cdot b = b^{1+1} = b^2$.

Combine the Results

Combine the new coefficient with the simplified variables: The expression is now $-16 a^{3} b^{2}$. This represents the fully simplified form of the initial expression.

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Key Concepts

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

Coefficients

When simplifying algebraic expressions, one of the key concepts is understanding and working with coefficients. Coefficients are the numbers placed in front of variables in an algebraic expression. In the expression given in the exercise, \[-8 a_2 b \cdot 2 a b\] the coefficients are $-8$ and $2$.To simplify the expression, you first need to multiply these coefficients together. This means performing a simple multiplication: $-8 \times 2 = -16$. The resulting coefficient, $-16$, becomes part of the simplified expression.

Coefficients define the numerical part of terms.
They are essential in arithmetic operations like addition, subtraction, or multiplication.
Multiplying coefficients is the first step in expression simplification.

Laws of Exponents

The laws of exponents play a crucial role when we multiply expressions involving powers. Exponents tell us how many times a number, known as the base, is multiplied by itself. These laws enable us to simplify expressions like the one in our exercise.Consider the part of the expression with variables: $a_2 b \cdot a b$. According to the laws of exponents, when we multiply terms that have the same base, we add their exponents together. This means:

$a_2 \cdot a = a^{2+1} = a^3$
$b \cdot b = b^{1+1} = b^2$

These laws make it possible to condense expressions, making them simpler and much easier to work with.

Exponents indicate repeated multiplication.
Add exponents when multiplying bases that are the same.
This helps to keep expressions compact and manageable.

Multiplying Expressions

Combining everything together, we can fully simplify the given expression by multiplying the coefficients and combining the variables using the laws of exponents. In our example, the original expression is $-8 a_2 b \cdot 2 a b$.First, we get the new coefficient by multiplying the numbers: $-16$.Next, apply the laws of exponents to multiply and simplify the variables:

$a_2 \cdot a = a^3$
$b \cdot b = b^2$

Finally, combine these results to form the simplified expression $-16 a^3 b^2$.This is how multiplying expressions involves systematically applying several rules:

Start with coefficients.
Use exponents laws on variables.
Combine all parts to reach the simplified form.

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91影视

Simplify. $$ -8 a_{2} b \cdot 2 a b $$

Short Answer

Step by step solution

Identify the Terms and Multiply Coefficients

Apply the Laws of Exponents to the Variables

Combine the Results

Key Concepts

Coefficients

Laws of Exponents

Multiplying Expressions

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