Problem 45 - College Algebra

Question

Simplify each exponential expression $$ \left(3 x^{4}\right)\left(2 x^{7}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(6x^{11}\).

Step 1: Multiply the coefficients

Multiply the coefficients \(3\) and \(2\) together to get \(6\).

Step 2: Add the exponents

Add the exponents together. So \(4 + 7 = 11\), this means \(x^{4+7}\) simplifies to \(x^{11}\).

Step 3: Write out the simplified expression

Multiply the resulting coefficient from step 1 with the simplified base from step 2 together to get the simplified expression. So we get \(6x^{11}\).

Key Concepts

Coefficients

In mathematics, coefficients play a crucial role in simplifying expressions. Coefficients are the numerical part of terms in an expression. In our exercise, we have coefficients 3 and 2 from the terms \(3x^4\) and \(2x^7\), respectively. When simplifying an expression that involves multiplication, it's essential to multiply these coefficients together first. This process simplifies the numerical part of the expression before dealing with the variable parts. For example, multiplying coefficients 3 and 2 gives us 6. Therefore, the product of the coefficients for the expression \((3x^4)(2x^7)\) is 6. This method keeps the calculations organized and straightforward.

Exponents

Exponents are a key component of exponential expressions, dictating how many times a number (the base) is multiplied by itself. In the expression \(x^4\), 4 is the exponent of base \(x\), meaning \(x\) is used as a factor 4 times (\(x \cdot x \cdot x \cdot x\)). In our exercise, we are given two exponents: 4 and 7.When multiplying terms with the same base, we add the exponents to simplify the expression: \(x^4 \cdot x^7\). This results in \(x^{4+7} = x^{11}\). Adding the exponents helps us express the repeated multiplication of a base more succinctly, making complex expressions more manageable. Remember, the rule of adding exponents only applies when the bases are identical.

Simplification

Simplification in mathematics involves reducing an expression to its simplest form. This often makes it easier to understand and solve further calculations. In our exercise, we simplify the expression \((3x^4)(2x^7)\) into a more concise form. Simplification combines the multiplication of coefficients and the addition of exponents.

• First, we multiply the coefficients: \(3 \times 2 = 6\).
• Then, we add the exponents of the similar base: \(4 + 7 = 11\), resulting in \(x^{11}\).

Bringing together these results, the simplified form of the expression is \(6x^{11}\). This approach prioritizes clarity and efficiency in solving expressions, making sure the overall computation is significantly reduced to its most basic form.