Problem 26 - Beginning and Intermediate Algebra

Question

Simplify. $$\left(8 a b^{7}\right)\left(-7 a^{-5} b^{2}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(-56a^{-4}b^{9}\).

Step 1: Identify the expressions to be multiplied

We will multiply the two given expressions: \(\left(8 a b^{7}\right) \) and \(\left(-7 a^{-5} b^{2}\right)\)

Step 2: Multiply the coefficients

Now, we will multiply the coefficients 8 and -7. The result will be the coefficient of the simplified product: \(8 \cdot (-7) = -56\)

Step 3: Apply the property of exponents for the variable a

Since both expressions have the variable a, we will add their exponents: \(a^{1} \cdot a^{-5} = a^{1 + (-5)} = a^{-4}\)

Step 4: Apply the property of exponents for the variable b

Similarly, since both expressions have the variable b, we will add their exponents: \(b^{7} \cdot b^{2} = b^{7 + 2} = b^{9}\)

Step 5: Combine the results

Now, we will combine the results from Steps 2, 3, and 4 to form the simplified expression: \(-56a^{-4}b^{9}\) So, the simplification of the given expression is \(-56a^{-4}b^{9}\).

Key Concepts

Properties of Exponents

The properties of exponents are essential rules that help us simplify expressions involving powers of the same base. When multiplying expressions with the same base, we can add the exponents together. This fundamental property makes it easier to handle complex algebraic expressions.

Let's explore how this works using our exercise:

• For the variable \(a\), the exponents are 1 and -5. When multiplying \(a^1\) and \(a^{-5}\), we add the exponents: \(1 + (-5) = -4\). Thus, \(a^1 \cdot a^{-5} = a^{-4}\).
• For the variable \(b\), the exponents are 7 and 2. We add these exponents: \(7 + 2 = 9\), resulting in \(b^7 \cdot b^2 = b^{9}\).

This addition property holds true for any real number exponents, streamlining the multiplication process and making expressions less bulky. Remember, this rule only applies when the base is the same.

Multiplication of Coefficients

The coefficients in an algebraic expression are the numerical parts that are directly multiplied together. To simplify an expression, multiplying the coefficients correctly is crucial. In our example, the coefficients are 8 and -7.

When you multiply these coefficients, you simply perform a regular multiplication, keeping in mind the rules for handling positive and negative numbers:

• Multiply the numbers: \(8 \times (-7)\).
• The product is -56, since a positive times a negative gives a negative result.

So when multiplying coefficients, treat them like regular numbers. Pay careful attention to their signs, as a negative product can change the entire expression’s meaning.

Simplification of Expressions

Simplifying expressions involves performing all possible operations to condense an expression into its simplest form. In algebra, this often means multiplying coefficients, applying exponent rules, and reorganizing terms.

For our initial expression \((8ab^7)(-7a^{-5}b^2)\), here's how we proceed:

• First, multiply the coefficients from each part, giving us -56.
• Next, apply properties of exponents to \(a\) and \(b\). For \(a\), combined exponents result in \(a^{-4}\). For \(b\), they total to \(b^9\).
• Finally, combine these results into a single expression: \(-56a^{-4}b^9\).

The simplified form is easier to interpret and often needed in more complex mathematical problems. Simplification aims to make the expression cleaner and sometimes reveals relationships not easily visible in its original form.