Problem 122 - Algebra and Trigonometry Real Mathematics, Real People

Question

Simplify the expression. $$(18 x)^{0}(4 x y)^{2}\left(3 x^{-1}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \( 48xy^{2} \).

Step 1: Simplify \( (18x)^{0} \)

Any number raised to the power of 0 equals to 1. Therefore, \( (18x)^{0} = 1 \).

Step 2: Simplify \( (4xy)^{2} \)

A number or variable raised to the power of 2 means that it is multiplied by itself. Therefore, \( (4xy)^{2} = 4^{2} \cdot x^{2} \cdot y^{2} = 16x^{2}y^{2} \).

Step 3: Simplify \( (3x^{-1}) \)

A variable raised to the power of -1 means the reciprocal of this variable is taken. Therefore, \( 3x^{-1} = 3/x \).

Step 4: Combine

Combine the simplified components of the first 3 steps by multiplying them together. The product will be the simplified form of the given expression. Therefore, \( 1 \cdot 16x^{2}y^{2} \cdot 3/x = 48x^{2}y^{2} / x = 48xy^{2} \).

Key Concepts

Exponent Rules

Exponent rules can seem tricky at first, but they're like friends once you get to know them. One key rule to remember is that any number or term raised to the zero power is always equal to one. This is expressed as \( a^0 = 1 \) for any nonzero \( a \). That's why in Step 1 of the solution, \((18x)^0\) becomes \(1\).
Another important rule involves multiplying exponents when you have a term raised to a power, such as \((4xy)^2\). Here, you apply the power to each component, turning the base, \(4xy\), into \(4^2 \cdot x^2 \cdot y^2\). This works because exponentiation distributes over multiplication.
Don't forget the rule for negative exponents: if you have a negative exponent, it implies the reciprocal of the term. For example, \(x^{-n}\) is the same as \(1/x^n\). This becomes useful when simplifying terms like \(3x^{-1}\), which turns into \(3/x\). With these rules, simplifying expressions with exponents becomes much simpler!

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operation symbols. These expressions must be simplified by following mathematical principles and rules.

• Variables and Constants: Variables represent unknown values, often seen as letters like \(x\) and \(y\). Constants are numbers on their own.
• Operations: They can include addition, subtraction, multiplication, division, and exponentiation, which states how many times to multiply a base by itself.

To simplify an expression, you need to perform all possible operations: combine like terms, use exponent rules, and apply distribution where necessary. For example, in \((4xy)^2\), applying each operation step-by-step ensures an accurate result, turning \(4xy\) into \(16x^2y^2\).
Ultimately, simplified expressions are easier to work with, whether you're solving equations or performing calculations.

Negative Exponents

Negative exponents might seem intimidating, but they're just a way of expressing division in a neat manner. When you see an exponent like \(x^{-1}\), it means the reciprocal, which flips the base upside down.

• If you have \(x^{-1}\), it turns into \(1/x\).
• Similarly, \(x^{-n}\) becomes \(1/x^n\), suggesting multiple divisions.

Understanding negative exponents is crucial for manipulating expressions accurately. They play a significant role in Step 3 of our solution, where \(3x^{-1}\) transforms into \(3/x\).
Recognizing how to handle negative exponents will boost your confidence in algebra. Instead of fear, think of them as invitations to uncover more truths about numbers and variables! By embracing them, you'll be well-equipped to tackle complex algebraic challenges effortlessly.