Problem 24 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ (\sqrt{16})^{4} $$

Step-by-Step Solution

Answer

The value of the expression \((\sqrt{16})^{4}\) is \(256\).

Step 1: Calculate the Square Root

The first step is to calculate the square root of \(16\). This is done by finding a number which, when multiplied by itself, gives the number \(16\). In this case, the square root of \(16\) is \(4\). So, the expression becomes \(4^{4}\).

Step 2: Use Exponent Rule

In the next step, it's necessary to raise the number \(4\) to the power of \(4\). This means multiplying \(4\) by itself four times. So, \(4^{4}\) equals to \(4*4*4*4\).

Step 3: Multiply the Values

By multipliying the values together: \(4*4=16\), then multiply \(16\) with \(4\) two more times: \(16*4=64\) and then \(64*4=256\). Hence, \(4^{4}=256\).

Key Concepts

Calculating Square Roots

Understanding how to calculate square roots is fundamental in algebra. A square root of a number is a value that, when multiplied by itself, equals the original number. For instance, the square root of 16 is 4 because when you multiply 4 by itself, which is noted as \(4 \times 4\), you get 16.

To evaluate square roots without a calculator, it helps to memorize the square roots of perfect squares like 1, 4, 9, 16, 25, and so on. In the exercise, recognizing \( 16 \) as a perfect square allows us to easily know that the square root of \(16\) is \(4\).

Exponent Rules

When dealing with exponents, it's important to remember the basic exponent rules that simplify these types of mathematical expressions. One key rule is that when a number is raised to a power, it's multiplied by itself as many times as the value of the exponent.

For example, \(4^4\) means you need to multiply \(4\) by itself four times: \(4 \times 4 \times 4 \times 4\). It's also useful to know that any number to the power of \(1\) is the number itself, and any number to the power of \(0\) is \(1\), excluding \(0\) which is undefined for the power of zero.

Multiplication of Values

Multiplying values effectively is another essential skill. In our exercise, after determining the square root of the number, we need to raise it to an exponent, resulting in a multiplication problem. Multiplication is a shortcut for repeated addition.

To multiply numbers, like in \(4^4\), you can start by multiplying the first two numbers to get \(16\), and then continue multiplying by \(4\) sequentially to reach the final answer.

It's also useful to remember multiplication tricks, such as anytime you multiply by 10, you simply add a zero to the end of the other number, or when multiplying by 5, you can halve the other number and then multiply by 10.