Evaluate square root of -3^2+-4^2
This question asks you to calculate the numerical value of an expression involving the square root of the sum of two terms: the square of negative three, and the negative of the square of four. The notation "-3^2" refers to the operation of squaring -3, which gives you the result of 9 (since squaring a negative number results in a positive number), and "-4^2" refers to taking the negative of the square of 4, which is also 16. Thus, you are asked to determine the square root of the sum of these two computed values.
$\sqrt{- 3^{2} + - 4^{2}}$
Answer
Solution:
Step 1:
Consolidate the $+$ and $-$ signs into a single $-$ sign. When a plus sign is immediately followed by a minus sign, it is equivalent to a minus sign alone, since $1 \cdot -1 = -1$. Thus, we have $\sqrt{-3^2 - 4^2}$.
Step 2:
Begin simplifying the given expression.
Step 2.1:
Calculate the square of $3$, which is $\sqrt{-1 \cdot 9 - 4^2}$.
Step 2.2:
Perform the multiplication of $-1$ with $9$, resulting in $\sqrt{-9 - 4^2}$.
Step 2.3:
Calculate the square of $4$, which is $\sqrt{-9 - 1 \cdot 16}$.
Step 2.4:
Perform the multiplication of $-1$ with $16$, resulting in $\sqrt{-9 - 16}$.
Step 2.5:
Combine $-9$ and $-16$ to get $\sqrt{-25}$.
Step 3:
Express $-25$ as the product of $-1$ and $25$, which is $\sqrt{-1(25)}$.
Step 4:
Separate the square root of the product into the product of square roots, $\sqrt{-1} \cdot \sqrt{25}$.
Step 5:
Replace the square root of $-1$ with the imaginary unit $i$, giving us $i \cdot \sqrt{25}$.
Step 6:
Express $25$ as the square of $5$, which is $i \cdot \sqrt{5^2}$.
Step 7:
Extract terms from under the square root, assuming they represent positive real numbers, to get $i \cdot 5$.
Step 8:
Rearrange the terms to place the numerical coefficient before the imaginary unit, resulting in $5i$.
Knowledge Notes:
The problem involves evaluating the square root of a sum of squares with negative signs. This is a common problem in complex number arithmetic. Here are the relevant knowledge points:
Combining Signs: The combination of a plus sign followed by a minus sign can be simplified to a single minus sign.
Exponentiation: Squaring a number means multiplying the number by itself.
Multiplication by -1: Multiplying a number by -1 changes its sign.
Square Root of a Negative Number: The square root of a negative number involves the imaginary unit $i$, where $i^2 = -1$.
Separation of Square Roots: The square root of a product can be expressed as the product of the square roots of the individual factors.
Extraction of Square Roots: When extracting the square root of a perfect square, the result is the positive number that, when squared, gives the original number.
Imaginary Numbers: Imaginary numbers are expressed as a real number multiplied by the imaginary unit $i$.
Order of Terms: In complex numbers, it is conventional to write the coefficient before the imaginary unit (e.g., $5i$ instead of $i5$).
In this problem, we use these concepts to simplify the expression step by step, ultimately arriving at a complex number in standard form.
