Question

Simplify each expression. \(\sqrt{(-4)^{2}}\)

Short answer

The simplified form is 4.

Step-by-step solution

- Identify the expression inside the square root

The expression given is \(\root{(-4)^{2}}\). First, recognize that the expression inside the square root is \((-4)^{2}\).

- Calculate the square of the number

Calculate \(-4^{2}\). To do this, multiply -4 by itself: \(-4\times-4=16\). Therefore, \((-4)^{2} = 16\).

- Simplify the square root

Now simplify the square root of 16. \(\root{16} = 4\). The square root of 16 is 4.

Key concepts

square roots

Square roots help us find a number that, when multiplied by itself, gives us the original number inside the root. For instance, if you have \(\root{36}\), this means finding a number which, when squared (or multiplied by itself), equals 36. The answer is 6, since \(6 \times 6 = 36\).
Common square roots you might encounter are:

• \(\root{4} = 2\)
• \(\root{9} = 3\)
• \(\root{16} = 4\)

Remember, square roots always produce a non-negative result. So, \(\root{16} = 4\), not -4, even though \((-4 \times -4 = 16)\) is true.

exponents

Exponents are shorthand for repeated multiplication. If you see an exponent like \(-4^{2}\), it means multiplying -4 by itself. So, \(-4 \times -4 = 16\). The most important rule here is to remember that if the base number (the number being multiplied) is negative, the result will depend on whether the exponent is even or odd:

• An even exponent (like 2, 4, 6) will give a positive result because multiplying two negatives makes a positive.
• An odd exponent (like 1, 3, 5) will give a negative result because you'll have an odd number of negative multiplications.

Thus, for \((-4)^2 = 16\), but \((-4)^3 = -64\)

basic algebra

Basic algebra involves dealing with variables and constants to solve equations and simplify expressions. When working with algebra, rules of operations (like Order of Operations, PEMDAS) and properties of numbers help us simplify expressions efficiently. In our example exercise, the algebraic operations we use include:

• Identifying the expression to understand the problem's components.
• Applying exponent rules to find the value of the number inside the square root.
• Simplifying the overall expression by taking the square root of the result.

Basic algebra keeps problems manageable by breaking them into small, simple steps.

simplification

Simplification is all about making an expression as easy as possible to work with, without changing its value. In our example, we first identified \((-4)²\), calculated it as 16, and then simplified the square root of 16 to get the final answer, 4. Remember these key points for simplification:

• Work inside parentheses and deal with exponents first.
• Move outwards: Simplify any square roots or other functions next.
• Continue simplifying until you have the simplest form possible.
Simplification helps in checking your work and ensures that equations or expressions are easier to handle in future steps or more complex problems.