Problem 47 - College Algebra Essentials

Question

Simplify each exponential expression. $$\left(-9 x^{3} y\right)\left(-2 x^{6} y^{4}\right)$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(18x^9y^5\).

Step 1: Simplify the Coefficient

First, multiply the coefficients which are -9 and -2. The result is \(-9 \times -2 = 18\). Remember that the product of two negatives is a positive.

Step 2: Apply the Law of Exponents for the 'x' Terms

When multiplying terms with the same base (in this case \(x\)), add the exponents. That means, add the exponents of \(x^3\) and \(x^6\) to get \(x^{3+6}\), which simplifies to \(x^9\).

Step 3: Apply the Law of Exponents for the 'y' Terms

In a similar way, add the exponents of \(y^1\) and \(y^4\) (consider the absence of an exponent on \(y\) in the first term as a 1) to get \(y^{1+4}\), which simplifies to \(y^5\).

Step 4: Combine All Parts

Combine all parts to get the simplified form of the given expression. This will be the coefficient from Step 1, the simplified 'x' term from Step 2, and the simplified 'y' term from Step 3, resulting in \(18x^9y^5\).