Problem 67 - Algebra 2

Question

Critical Thinking When \(\sqrt{x^{a} y^{b}}\) is simplified, the result is \(\frac{1}{x^{c} y^{3 d}}\) , where \(c\) and \(d\) are positive integers. Express \(a\) in terms of \(c,\) and \(b\) in terms of \(d .\)

Step-by-Step Solution

Answer

The value of \(a\) in terms of \(c\) is \(a=-2c\) and the value of \(b\) in terms of \(d\) is \(b=-6d\).

Step 1: Simplification of square root

The expression under the square root \(\sqrt{x^{a} y^{b}}\) can be simplified as \(x^{\frac{a}{2}} y^{\frac{b}{2}}\). According to the rules of exponents, taking the square root of a power is equal to halving the exponent.

Step 2: Simplification of the result

The result is given as \(\frac{1}{x^{c} y^{3d}}\), which is equivalent to \(x^{-c} y^{-3d}\). When a base with its exponent is in the denominator, it is equivalent to the base with its exponent being negative.

Step 3: Equate and solve for \(a\) and \(b\)

Now setting \(x^{\frac{a}{2}} y^{\frac{b}{2}}\) equal to \(x^{-c} y^{-3d}\), we get two equations. For the bases to be equal, the exponents must also be equal. Therefore, \(\frac{a}{2} = -c\) and \(\frac{b}{2} = -3d\). Solving these for \(a\) and \(b\), we get \(a = -2c\) and \(b = -6d\).

Key Concepts

Exponents

Exponents are a vital part of algebra that indicate how many times a number, known as the base, is multiplied by itself. For example, in the expression \(x^a\), \(x\) is the base and \(a\) is the exponent. This expression tells us to multiply \(x\) by itself \(a\) times.

• If \(a = 2\), the expression \(x^2\) means \(x\cdot x\).
• If \(a = 3\), then \(x^3 = x \cdot x \cdot x\).

Understanding and using exponents is essential for simplifying algebraic expressions and solving equations in mathematics.
In the context of the exercise, we have exponents \(a\) and \(b\) attached to variables \(x\) and \(y\) respectively. These need to be manipulated through various rules of exponents to solve the equation.

Square Roots

The square root function is the opposite of squaring a number. In mathematics, the square root of a number \(n\) is a value \(m\) that, when multiplied by itself, equals \(n\). This is denoted by \(\sqrt{n}\).

• For example, \(\sqrt{9} = 3\), because \(3 \times 3 = 9\).
• Similarly, \(\sqrt{16} = 4\).

For algebraic expressions like \(\sqrt{x^a y^b}\), the square root affects each of the terms inside.
This means you apply the square root to each part: \(x^a\) becomes \(x^{a/2}\), and \(y^b\) becomes \(y^{b/2}\). These principles are crucial for solving expressions where you need to "undo" the square or simplify expressions that involve roots.

Simplification

Simplification in algebra is the process of reducing expressions into a simpler form. This does not change the value of the expression; it only makes it easier to work with. Simplification often involves combining like terms, reducing fractions, and applying algebraic identities.
In our exercise, simplification is required to transform the square root expression \(\sqrt{x^a y^b}\) into a more workable form.

• We simplify \(\sqrt{x^a y^b}\) into \(x^{a/2} y^{b/2}\).
• The expression \(\frac{1}{x^c y^{3d}}\) simplifies to \(x^{-c} y^{-3d}\) by moving the terms in the denominator to the numerator.

Simplification allows us to equate terms and solve for unknown variables more effectively, as seen when matching coefficients of like bases.

Exponent Rules

Exponent Rules are a set of guidelines that help manage and manipulate expressions with exponents, making them easier to work with. Here are some key rules:

• Product of Powers: \(x^a \times x^b = x^{a+b}\).
• Quotient of Powers: \(\frac{x^a}{x^b} = x^{a-b}\).
• Power of a Power: \((x^a)^b = x^{a\times b}\).
• Negative Exponent: \(x^{-a} = \frac{1}{x^a}\).

These rules greatly assist in simplifying expressions, especially when working with equations involving multiple bases and exponents.
For this problem, the negative exponent rule is applied when expressing \(\frac{1}{x^c}\) as \(x^{-c}\). This is crucial in aligning the expression \(x^{a/2} = x^{-c}\) to find the relationship between \(a\) and \(c\). Similarly, using these rules helps relate \(b\) to \(d\) in the given exercise.