Problem 41 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ \left(9 x y^{3}\right)^{0} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression using power rules for exponents: (9xy^3)^0. Answer: 1

Step 1: Recall the power rules for exponents

The basic power rules for exponents are: 1. Any nonzero number raised to the power of 0 is 1. 2. When taking the power of a product, distribute the exponent to each factor in the product.

Step 2: Apply the power rule for exponent 0 to the given expression

Given expression: \(\left(9 x y^{3}\right)^{0}\) According to the power rule for exponent 0, the given expression simplifies to 1, since all bases are nonzero: $$ \left(9 x y^{3}\right)^{0} = 1 $$

Step 3: Write the simplified expression

The simplified expression is: $$ 1 $$

Key Concepts

Exponentiation

Exponentiation is a mathematical operation, written as a^n, involving two numbers, the base a and the exponent n. When n is a natural number, exponentiation corresponds to repeated multiplication of the base: that is, a^n is the product of multiplying a by itself n times.

For example, 3^4 (read as 'three to the fourth power' or 'three to the power of four') is 3 × 3 × 3 × 3, which equals 81. This operation is fundamental in algebra and occurs frequently in various mathematical scenarios, such as growth processes and area calculations.

To simplify an expression with exponentiation, one must understand and apply the rules or properties governing this operation, which include the power rules for exponents as shown in our exercise problem.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing the expression to its most basic form without changing its value. It's like tidying up a cluttered room so that it's easier to understand and work within.

To simplify an expression with exponents, it's essential to know the specific rules that apply. One of these rules comes into play when an entire expression raised to the zero power is involved, as in our textbook example. Recognizing that any nonzero base raised to the power of zero equals one can immediately simplify the expression to its simplest form, which is 1.

This rule is particularly handy for dealing with large or complex algebraic expressions, as it can greatly reduce the effort needed for further calculations. Remember, simplifying expressions is not about altering their inherent value—it's about making them easier to interact with.

Properties of Exponents

The properties of exponents are the rules that guide the simplification process when working with exponentiation. These properties are crucial for correctly manipulating algebraic expressions involving powers.

• Zero Exponent Rule: If the exponent is 0, the expression equals 1, as long as the base is not 0. This is why (9xy^3)^0 simplifies to 1.
• Product of Powers Rule: When you're multiplying two powers that have the same base, you add the exponents. For instance, x^m * x^n = x^(m+n).
• Power of a Power Rule: To find a power of a power, you multiply the exponents. For example, (x^m)^n = x^(m*n).
• Power of a Product Rule: When raising a product to a power, you raise each factor to the power separately: (ab)^n = a^n * b^n.
• Negative Exponent Rule: A negative exponent indicates division by the base raised to the corresponding positive exponent: x^-n = 1/x^n.

Mastering these properties is fundamental for any student's success in algebra. They not only aid in simplification but also in understanding the underlying concepts of how numbers and variables operate within equations.