Simplify -5 cube root of -64+ cube root of -1- square root of -25
The problem provided involves simplifying a mathematical expression that contains cube roots and square roots of negative numbers, which suggests the use of imaginary numbers. The expression consists of subtracting the cube root of -1 and the square root of -25 from the cube of the cube root of -64. The objective is to perform these operations by applying the properties of exponents and roots, as well as the concept of imaginary numbers, to express the final result in its simplest form.
$- 5 \sqrt[3]{- 64} + \sqrt[3]{- 1} - \sqrt{- 25}$
Answer
Solution:
Step:1
Break down each term separately.
Step:1.1
Express $-64$ as $(-4)^3$.
$-5\sqrt[3]{(-4)^3} + \sqrt[3]{-1} - \sqrt{-25}$
Step:1.2
Extract terms from under the cube root, where all numbers are real.
$-5 \cdot -4 + \sqrt[3]{-1} - \sqrt{-25}$
Step:1.3
Calculate the product of $-5$ and $-4$.
$20 + \sqrt[3]{-1} - \sqrt{-25}$
Step:1.4
Represent $-1$ as $(-1)^3$.
$20 + \sqrt[3]{(-1)^3} - \sqrt{-25}$
Step:1.5
Extract terms from under the cube root, where all numbers are real.
$20 - 1 - \sqrt{-25}$
Step:1.6
Rewrite $-25$ as $-1(25)$.
$20 - 1 - \sqrt{-1(25)}$
Step:1.7
Separate the square root of the product into the product of square roots.
$20 - 1 - (\sqrt{-1} \cdot \sqrt{25})$
Step:1.8
Replace $\sqrt{-1}$ with the imaginary unit $i$.
$20 - 1 - (i \cdot \sqrt{25})$
Step:1.9
Express $25$ as $5^2$.
$20 - 1 - (i \cdot \sqrt{5^2})$
Step:1.10
Extract terms from under the square root, where all numbers are positive real numbers.
$20 - 1 - (i \cdot 5)$
Step:1.11
Position the $5$ before the imaginary unit $i$.
$20 - 1 - (5 \cdot i)$
Step:1.12
Combine the real numbers.
$20 - 1 - 5i$ $19 - 5i$
Step:2
Subtract $1$ from $20$ to get the final result.
$19 - 5i$
Knowledge Notes:
The problem involves simplifying an expression that contains cube roots, square roots, and imaginary numbers. Here are the relevant knowledge points:
Cube Root: The cube root of a number $x$ is a number $a$ such that $a^3 = x$. The cube root of a negative number is also negative.
Square Root: The square root of a number $x$ is a number $a$ such that $a^2 = x$. The square root of a negative number is not a real number but an imaginary number.
Imaginary Numbers: Imaginary numbers are an extension of the real number system where the square root of $-1$ is denoted as $i$. Thus, $\sqrt{-1} = i$.
Simplifying Radical Expressions: When simplifying expressions involving roots, it's often helpful to rewrite numbers as powers that match the index of the root. For example, $64$ can be written as $4^3$ when dealing with a cube root.
Combining Like Terms: When simplifying expressions, combine like terms to simplify the expression to the greatest extent possible.
Real Numbers: In this context, when we assume all numbers are real, we are considering numbers without the imaginary unit $i$.
Multiplication of Imaginary Numbers: When multiplying an imaginary number by a real number, the real number is usually written first. For example, $5i$ is the product of $5$ and $i$.
Simplifying Expressions with Imaginary Numbers: To simplify expressions with imaginary numbers, we often separate the real and imaginary parts and simplify them separately.
By applying these principles, we can simplify the given expression step by step, ensuring that each radical and imaginary number is handled correctly.
