Problem 48 - Algebra for College Students

Question

Raise each monomial to the indicated power. $$\left(8 x^{2} y^{5}\right)^{2}$$

Step-by-Step Solution

Answer

The expression \((8x^2y^5)^2\) simplifies to \(64x^4y^{10}\).

Step 1: Understanding the Problem

The given expression is \((8x^2y^5)^2\). Our task is to raise each component of the monomial \(8x^2y^5\) to the power of 2.

Step 2: Applying the Power Rule for Exponents

The power rule for exponents states that \((a^m)^n = a^{m \times n}\). We apply this rule to each component of the monomial \(8x^2y^5\).

Step 3: Raising the Coefficient to the Power

Raise the coefficient 8 to the power of 2: \(8^2 = 64\).

Step 4: Raising the \(x\) Term to the Power

For the term \(x^2\), apply the power rule: \((x^2)^2 = x^{2 \times 2} = x^4\).

Step 5: Raising the \(y\) Term to the Power

For the term \(y^5\), apply the power rule: \((y^5)^2 = y^{5 \times 2} = y^{10}\).

Step 6: Constructing the Final Expression

Combine all the raised components to get the final expression: \(64x^4y^{10}\).

Key Concepts

Power Rule for Exponents

The power rule for exponents is a fundamental concept in algebra that helps us simplify expressions involving powers raised to another power. When you have an expression such as \((a^m)^n\), you can simplify it using the power rule, which states that this is equal to:\[a^{m \times n}\]where \(a\) is the base and \(m\) and \(n\) are the exponents.

This means you multiply the exponents together.
For each base, whatever the power is, you "multiply" that power to the outer power.
Take for example \((x^2)^2\): According to the power rule, it becomes\[x^{2 \times 2} = x^4\].

The power rule is very useful when dealing with monomials that are in brackets raised to a power.

• Ensure each term inside the brackets is clearly understood.
• Apply the power law with accurate calculations to fully simplify the monomial to a new, much simpler form.

Exponentiation

Exponentiation is the process of raising a number (or a variable) to a given power.
This means multiplying the base by itself a certain number of times, which is specified by the exponent.
For example, \(8^2\) signifies that the number 8 is to be multiplied by itself: \(8 \times 8 = 64\).

When the base is numeric, such as 8, you can compute the result directly.
However, if the base is an algebraic expression such as \(x^p\), then exponentiation involves applying the power rule for exponents.
This multiplication process of the exponents makes organizing, arranging, and reducing expressions far more manageable.

• Helps in simplifying expressions and making calculations easier.
• Allows for the expansion and reduction of larger algebraic expressions efficiently.

It’s a fundamental operation in mathematics and a key component in many algebraic calculations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations put together to represent real-world quantities and solve various problems.
Some key characteristics of algebraic expressions include:

• Terms - consists of numbers and variables multiplied together.
• Operators - like addition, subtraction, multiplication, and division.
• Coefficients - numerical parts of the terms.
• Variables - symbols, often "x" or "y", that represent unknown quantities.

In the expression \(8x^2y^5\), 8 is the coefficient, x and y are variables, and 2 and 5 are exponents attached to those variables.

Monomials, like \(8x^2y^5\), are a type of algebraic expression that have only one term.
Recognizing these components is vital for manipulating and solving algebraic expression problems.
They help to frame how equations can be simplified or expanded based on specific rules like the power rule for exponents.
Understanding and working with algebraic expressions is critical in mathematics as they form the foundation of more complex calculations and problem-solving scenarios.