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Magazine Chapter 11: Problem 19
Evaluate the given expressions. $$\frac{1000^{1 / 3}}{-400^{-1 / 2}}$$
Short Answer
Expert verified
The expression evaluates to -200.
Step by step solution
01
Understand the Expression
The task is to evaluate the expression \( \frac{1000^{1 / 3}}{-400^{-1 / 2}} \). This involves finding the cube root and the reciprocal square root of numbers.
02
Calculate the Numerator
The numerator is \( 1000^{1/3} \). We need to find the cube root of 1000. Since \( 1000 = 10^3 \), we have \( 1000^{1/3} = (10^3)^{1/3} = 10^{3/3} = 10 \).
03
Calculate the Denominator
The denominator is \( -400^{-1/2} \). The term \( 400^{-1/2} \) means the reciprocal of the square root of 400. Since \( 400 = 20^2 \), \( \sqrt{400} = 20 \). Thus, \( 400^{-1/2} = \frac{1}{\sqrt{400}} = \frac{1}{20} \). Therefore, \( -400^{-1/2} = -\frac{1}{20} \).
04
Simplify the Fraction
Now, substitute the values from Steps 2 and 3 into the fraction: \( \frac{10}{-\frac{1}{20}} \). Dividing by a fraction is equivalent to multiplying by its reciprocal, so this becomes \( 10 \times (-20) = -200 \).
05
Write the Final Result
The final result of the given expression is \( -200 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponentiation
Exponentiation is the mathematical operation where a number, known as the base, is raised to the power of an exponent. To visualize, if you have a base number like 2 and an exponent of 3, expressed as \( 2^3 \), it means you multiply the base 2 by itself three times: \( 2 \times 2 \times 2 = 8 \).
This process allows us to express large numbers succinctly and to perform operations on them efficiently.
This process allows us to express large numbers succinctly and to perform operations on them efficiently.
- Base: The number that is repeatedly multiplied.
- Exponent: Indicates how many times the base is used in the multiplication.
- Power: The result of exponentiation.
Cube root
The cube root of a number is a value that, when multiplied by itself three times, gives the original number. In mathematical notation, the cube root of a number \( x \) is written as \( x^{1/3} \). For example, the cube root of 1000 is 10 because \( 10 \times 10 \times 10 = 1000 \).
Finding a cube root is essentially reversing the process of cubing a number.
Finding a cube root is essentially reversing the process of cubing a number.
- If \( x = n^3 \), then \( n = x^{1/3} \).
- Cubes of whole numbers are precise, making cube roots for such numbers simple to compute.
Reciprocal
A reciprocal is what you multiply a number by to get 1. For any number \( a \), its reciprocal is \( \frac{1}{a} \). A simple example is the reciprocal of 5, which is \( \frac{1}{5} \). When you multiply 5 by \( \frac{1}{5} \), the result is 1.
Reciprocals are significant in division and solving equations.
Reciprocals are significant in division and solving equations.
- Reciprocal of a fraction: If you have \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
- Finding reciprocals helps simplify division into multiplication. Dividing by a number is the same as multiplying by its reciprocal.
Square root
A square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \). It is represented as \( \sqrt{x} \). For instance, \( \sqrt{400} = 20 \) because \( 20 \times 20 = 400 \).
Your calculator can quickly find square roots, and they are very important in various equations and scientific calculations.
Your calculator can quickly find square roots, and they are very important in various equations and scientific calculations.
- Perfect Squares: Numbers like 1, 4, 9, 16, 25, etc., have whole number square roots.
- Principle of Square Roots: Every positive number has two square roots: one positive and one negative. \( \sqrt{400} = 20 \) and \( \sqrt{400} = -20 \).
