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Chapter 0: Problem 112

Simplify each expression. Assume that all variables represent positive numbers. $$\left(8 x^{-6} y^{3}\right)^{\frac{1}{3}}\left(x^{\frac{5}{6}} y^{-\frac{1}{3}}\right)^{6}$$

Short Answer

Expert verified
The simplified form of the given expression is \(2x^{7} y^{3}\)

Step by step solution

01

Simplify the Inside of the Parentheses

Find the cube root of each term inside the parentheses. The cube root of 8 is \(2\), and the cube root of \(x^{-6}\) is \(x^{-2}\) since \(-6 梅 3 = -2\). Similarly, the cube root of \(y^{3}\) is \(y\). The left parenthesis simplifies to: \( 2 x^{-2} y \).\nNow, we will simplify the terms inside the second parentheses. The exponent to which the inside terms of the parentheses is raised changes each individual exponent inside. So \(x^{\frac{5}{6}}\) to the 6th power becomes \(x^{5}\), and \(y^{-\frac{1}{3}}\) to the 6th power becomes \(y^{-2}\)
02

Multiply the Simplified Parentheses

Now that both parentheses have been simplified, they can be multiplied together. When multiplying terms with the same base, the exponents are added together.\nThe term for `x` becomes: \(x^{-2} \cdot x^{5} = x^{5-(-2)} = x^{7}\)\nThe term for `y` becomes: \(y \cdot y^{-2} = y^{1-(-2)} = y^{3}\)
03

Resolving the Final Simplified Form

After combining like terms, our final answer simplifies to: \(2x^{7} y^{3}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
Simplifying expressions involves reducing a complex mathematical expression into its simplest form. This makes the expression easier to work with and understand.
  • Identify and combine like terms.
  • Use fundamental math operations, such as multiplication and exponentiation.
  • Apply mathematical properties, such as the distributive and associative laws.
Breaking down expressions into manageable parts is key. Start by dealing with operations inside parentheses and follow the order of operations鈥攐ften remembered as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). By simplifying each part gradually, we reduce the complexity of the expression, leading to a clearer and more concise solution.
Exponents
Exponents are a shorthand way of multiplying a number by itself multiple times. The number being multiplied is called the base, and the exponent indicates how many times the base is used as a factor.
  • For example, the expression \(x^3\) means \(x\) multiplied by itself twice: \(x \times x \times x\).
  • Negative exponents indicate reciprocal; for example, \(x^{-3} = \frac{1}{x^3}\).
When dealing with exponents, remember these key rules:
  • When multiplying like bases, add the exponents: \(a^m \times a^n = a^{m+n}\).
  • When raising a power to another power, multiply the exponents: \((a^m)^n = a^{m \times n}\).
Understanding and applying these rules properly helps in transforming and simplifying expressions involving powers.
Cube Root
The cube root of a number is a value that, when multiplied by itself three times, returns the original number. It is related to exponents and is denoted using the radical symbol with a small 3 above it: \(\sqrt[3]{x}\).
  • For example, the cube root of 8 is 2, because \(2 \times 2 \times 2 = 8\).
  • When dealing with expressions like \((8x^{-6}y^{3})^{\frac{1}{3}}\), separate the cube root operation across each factor:
    • Cube root of 8 yields 2.
    • Cube root of \(x^{-6}\) results in \(x^{-2}\), because dividing the exponent by 3 gives \(-2\).
    • Cube root of \(y^3\) is \(y\).
Mastering cube roots allows you to simplify radical expressions effectively, leading to more manageable algebraic manipulations.
Algebraic Manipulation
Algebraic manipulation includes a variety of techniques used to rearrange and simplify algebraic expressions. It is an essential skill in precalculus.
  • It involves applying operations like addition, subtraction, multiplication, and division.
  • Each operation must respect algebraic properties, such as distributive, associative, and commutative laws.
Here's the basic approach:
  • Combine like terms to consolidate the expression.
  • Apply the laws of exponents to handle powers of variables.
  • Use inverse operations to isolate variables if necessary.
For instance, in the expression \(2x^{-2}y \cdot x^{5}y^{-2}\), you can use the power addition rule to simplify it as \(2x^{7}y^{3}\). This simplification requires recognizing like terms and applying the rules systematically to achieve the final answer. Easy algebraic manipulation is fundamental for dealing with more complex problems, eventually making them more approachable and solvable.

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