Problem 95 - Intermediate Algebra

Question

Simplify each exponential expression. $$ \left(-3 x^{2} y^{3} z^{5}\right)\left(20 x^{5} y^{7}\right) $$

Step-by-Step Solution

Answer

-60x^7y^{10}z^5

Step 1: Distribute the Constants

Multiply the numeric coefficients: \(-3 \times 20 = -60\)

Step 2: Simplify the Exponents of x

Add the exponents for \(x\) from both expressions.\(-3x^2y^3z^5\) has \(x^2\) and \(20x^5y^7\) has \(x^5\):\(x^2 \times x^5 = x^{2+5} = x^7\)

Step 3: Simplify the Exponents of y

Add the exponents for \(y\) from both expressions. \(-3x^2y^3z^5\) has \(y^3\) and \(20x^5y^7\) has \(y^7\):\(y^3 \times y^7 = y^{3+7} = y^{10}\)

Step 4: Simplify the Exponents of z

Since \(z\) only appears in \(-3x^2y^3z^5\), its exponent remains unchanged:\(z^5\)

Step 5: Write the Final Expression

Combine the results of the previous steps to write the simplified expression, which is:\(-60 x^7 y^{10} z^5\)

Key Concepts

Multiplication of Polynomials

When multiplying polynomials, it's important to keep track of both the coefficients (the numbers in front of variables) and the variables themselves. For example, in the expression \((-3 x^{2} y^{3} z^{5})(20 x^{5} y^{7})\), you have two separate polynomial terms to combine. Start by multiplying the coefficients of each polynomial term:

• The coefficient from the first polynomial is \(-3\).
• The coefficient from the second polynomial is \(20\).

To find the product of these coefficients, multiply them directly: \(-3 \times 20 = -60\). This product becomes the new coefficient of the resulting polynomial. The process doesn't stop here; next, you need to account for the variables, which are handled by the rules of exponents properties. Remember:

• Each multiplication is separate but related.
• The coefficients are multiplied independently of the variables.
• Ignore variables when multiplying the numbers and focus on arranging the product of numbers first. This allows for organized simplification of the exponent parts next.

Exponents Properties

Exponents properties are essential when dealing with polynomial multiplication. A key rule to remember is that when you multiply like bases, you add their exponents. For instance, the expression contains variables with exponents, such as \(x, y,\) and \(z\).

• For \(x\), the exponents are \(2\) and \(5\). According to the properties of exponents, when you multiply \(x^2 \times x^5\), you add the exponents: \(x^{2+5} = x^7\).
• Similarly for \(y\), you combine \(y^3\) and \(y^7\) by adding their exponents: \(y^{3+7} = y^{10}\).
• The exponent of \(z\) remains \(z^5\) since there's only one term containing it in the multiplication.

Adding exponents is crucial for simplifying expressions effectively. It's much like working with sums, where you operate separately yet combine results by carefully adding up exponents that share the same base variable.

Algebraic Simplification

Algebraic simplification brings together everything you've worked on, combining coefficients and exponents into a single, simpler expression. Following the steps from both multiplication and properties of exponents, you can reconstruct the expression.Here's how it looks with our example:

• The coefficient calculated from the initial step is \(-60\).
• The simplified exponents for each variable were calculated as \(x^7\), \(y^{10}\), and \(z^5\).

Combining these results gives you the final expression: \(-60 x^7 y^{10} z^5\). This represents the combination of all terms, thoroughly simplified for any further algebraic operations or assessments.Remember, simplifying algebraic expressions to their most concise form is not only satisfying but essential for further math operations. It’s like solving a puzzle where every piece—coefficients and exponents—fits perfectly to form a clear picture of the expression.