Question

Simplify the expression. $$4 \sqrt{\frac{16}{4}}$$

Short answer

The simplified form of the expression is 8.

Step-by-step solution

Simplify the Division

The first step is to simplify the fraction under the square root. That is, \(\frac{16}{4}\) simplifies to 4.

Perform Square Root Operation

The next step is to perform the square root of 4 which gives 2.

Multiply by 4

Finally, remember there is a 4 outside of the square root. So the final step is to multiply 4 times the result of the square root operation to obtain 8.

Key concepts

Radical Expressions

When we talk about radical expressions, we're looking at mathematical expressions that include a radical symbol—typically a square root, but radicals can also represent cube roots, fourth roots, and so on. The radical expression essentially indicates that we're seeking the 'root' of a number which is the inverse operation of raising a number to a power. For instance, the square root of a number is a value that, when multiplied by itself, gives the original number.

In the given exercise \(4 \sqrt{\frac{16}{4}}\), we have a radical expression that includes a square root. Simplifying radical expressions usually involves simplifying the number under the radical sign to its most reduced form and then taking the appropriate root. It's also common to have coefficients, numbers outside the radical which multiply the expression, as in our example, where 4 is the coefficient multiplying the square root.

Simplifying Fractions

A fundamental skill when working with radical expressions, especially when they involve fractions, is the ability to simplify fractions. Simplifying a fraction means reducing it to its simplest form where the numerator and the denominator have no common factors other than 1. This is achieved by dividing both the numerator and the denominator by their greatest common divisor.

Simplifying the fraction \(\frac{16}{4}\) to 4, as done in the first step of our exercise, demonstrates this process. The number 16 divided by 4 results in the whole number 4, indicating that 16 and 4 share a common divisor, which in this case, is 4 itself.

Square Roots

Understanding the concept of square roots is crucial in mathematics. A square root asks the question: 'What number times itself equals the given number?' For example, the square root of 9 is 3 because 3 times 3 equals 9. When it comes to simplifying square root expressions, the goal is to find the square root of the number under the radical as efficiently as possible.

If the number is a perfect square—like 4, 9, 16, and so on—the square root will be a whole number. However, if the number is not a perfect square, the square root may be an irrational number. In our exercise, finding the square root of 4 in the second step leads us directly to the whole number 2.

Arithmetic Operations

Arithmetic operations are the backbone of simplifying radical expressions, including addition, subtraction, multiplication, and division. These operations are the tools we use to manipulate and combine numbers and variables.

In the context of this exercise, after simplifying the fraction and finding the square root, the final step involves multiplication—an arithmetic operation. We multiply the coefficient 4 by the square root of 4 (which is already simplified to 2) to yield the final answer of 8. This demonstrates how arithmetic operations are combined with simplification tactics to solve expressions involving radicals.