Question

Multiply and simplify. $$\left(15.9 x^{2}\right)\left(4.93 x^{4}\right)$$

Short answer

The simplified result is 78.387x^6.

Step-by-step solution

Multiply the Coefficients

First, multiply the numerical coefficients (constants) of the two terms. In this case, multiply 15.9 and 4.93.

Apply Exponent Rules

After multiplying the coefficients, apply the power rule for exponents which states that when multiplying like bases, you add the exponents. So, add the exponents of the variables: the exponent 2 from the first term and exponent 4 from the second term, both for the base x.

Combine the Multiplication Results

Combine the results from Step 1 and Step 2 to find the product of the original expression.

Key concepts

Power Rule for Exponents

Understanding the power rule for exponents is crucial when working with algebraic expressions. This rule is relatively simple: when multiplying two expressions with the same base, you keep the base and add the exponents. For instance, take the expression \(x^2 \cdot x^4\). Both parts of the multiplication have the same base, \(x\), and therefore, we can apply the power rule. You would add the exponents 2 and 4 together, resulting in \(x^6\).

This is an intuitive process once you think of exponents as representing how many times a number is multiplied by itself. Multiplication is associative, meaning you can group numbers in different ways and still get the same product. So \(x^2 \cdot x^4\) is the same as \(x \cdot x \cdot x \cdot x \cdot x \cdot x\) or \(x^6\).

Multiplying Coefficients

In algebra, coefficients are the numerical parts of terms. When you multiply terms with coefficients, you multiply the coefficients separately from the variables. The example from our textbook problem involves multiplying 15.9 and 4.93. You treat these numbers as you would in any basic math multiplication. It’s like a separate task from dealing with the variables. After you calculate the product of the numerical coefficients, you'll combine this result with the results from applying exponent rules to the variables.

It's important not to mix up coefficients with variables. They are like different species of mathematical elements: one is numerical and constant, and the other is variable and represents quantities that can change.

Algebraic Multiplication

Combining coefficients and variables is a common task in algebra called algebraic multiplication. When we multiply algebraic expressions, we do this in two parts: first, we multiply the coefficients, as we mentioned earlier. Then, we multiply the variables, often using exponent rules.

For instance, when multiplying \(15.9x^2\) and \(4.93x^4\), you deal with the coefficients 15.9 and 4.93 separately from the variable parts \(x^2\) and \(x^4\). After each part has been multiplied, you combine them to get the final product. Remember, the key is to keep the operations on numbers and variables apart until each step is completed.

Exponent Addition

Exponent addition is part of the power rule for exponents and is a specific type of algebraic multiplication where the bases of the exponents are the same. When you multiply expressions like \(a^m \cdot a^n\), you simply add the exponents: \((m + n)\) while keeping the base \((a)\) unchanged. This results in the expression \(a^{m+n}\).

Returning to our exercise, after finding the new coefficient by multiplying 15.9 by 4.93, you then look at the exponents of \(x\). Adding the exponents 2 and 4 gives us \(x^{2+4}\) or \(x^6\). This is how you use exponent addition to simplify an expression with a common base being raised to multiple powers through multiplication.