Equivalent Expressions To √40: Find The Right Answers

Hey guys! Let's dive into the world of square roots and equivalent expressions. This is a super important concept in mathematics, and understanding it can really boost your problem-solving skills. Today, we're tackling the expression $\sqrt{40}$ and figuring out which other expressions are just different ways of writing the same value. Think of it like this: we're searching for the secret identities of $\sqrt{40}$! So, grab your pencils, and let's get started!

Breaking Down √40: The Basics

First off, what does $\sqrt{40}$ even mean? The square root of a number is a value that, when multiplied by itself, gives you that original number. In this case, we're looking for a number that, when squared, equals 40. But 40 isn't a perfect square (like 9, 16, or 25), so its square root is an irrational number, meaning it can't be expressed as a simple fraction. That's where simplifying and finding equivalent expressions come in handy! So, we need to look for factors of 40 that are perfect squares. What does that look like? Well, let's break down the number 40 to its prime factors. This is a crucial step, guys, because it helps us see the perfect squares hiding within.

The prime factorization of 40 is $2 \times 2 \times 2 \times 5$, which can be written as $2^3 \times 5$. Now, remember that a square root "undoes" a square. So, if we can find pairs of the same prime factor, we can pull them out of the square root. We have a pair of 2s here ($2 \times 2$), which is $2^2$, or 4. That's a perfect square! So, we can rewrite $\sqrt{40}$ as $\sqrt{4 \times 10}$. And because the square root of a product is the product of the square roots (i.e., $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$), we get $\sqrt{4} \times \sqrt{10}$. We all know the square root of 4 is 2, so we can simplify this to $2\sqrt{10}$. There you go! We've just found one of the equivalent expressions. But the fun doesn't stop there; let's explore the other options to see if they match up.

Checking the Options: Which Expressions Match √40?

Now that we've simplified $\sqrt{40}$ to $2\sqrt{10}$, let's evaluate the given options and see which ones are equivalent. This is where we put on our detective hats and carefully compare each expression to our simplified form. Remember, equivalent expressions might look different, but they represent the same numerical value. So, we're essentially looking for different disguises of the same number.

Option 1: $4\sqrt{10}$

Is $4\sqrt{10}$ equivalent to $2\sqrt{10}$? Absolutely not! The coefficient in front of the square root is different. $4\sqrt{10}$ is twice as large as $2\sqrt{10}$. Think of it like having 4 bags with $\sqrt{10}$ apples each versus having only 2 bags with the same amount of apples. Clearly, 4 bags contain more apples than 2 bags. So, we can immediately rule out this option. It's a common mistake to confuse coefficients, so always double-check those numbers!

Option 2: $2\sqrt{10}$

This is the simplified form we derived earlier! So, yes, $2\sqrt{10}$ is indeed equivalent to $\sqrt{40}$. Give yourself a pat on the back if you spotted this one right away! It's important to recognize that simplifying a radical expression can lead you directly to its equivalent forms. This skill is super handy when dealing with more complex equations and problems. Keep practicing your simplification techniques, and you'll become a pro at spotting these matches.

Option 3: $40^{\frac{1}{2}}$

This one might look a little different, but it's actually a sneaky way of writing a square root. Remember, an exponent of $\frac{1}{2}$ is just another way of saying "take the square root." So, $40^{\frac{1}{2}}$ is exactly the same as $\sqrt{40}$. They're just wearing different outfits! This is a key concept to remember when dealing with exponents and radicals. Being able to switch between these forms can make solving problems much easier. So, this option is also equivalent!

Option 4: $160^{\frac{1}{2}}$

Let's think about this one. $160^{\frac{1}{2}}$ is the same as $\sqrt{160}$. Is $\sqrt{160}$ the same as $\sqrt{40}$? Nope! 160 is much larger than 40, so its square root will also be larger. We can also simplify $\sqrt{160}$ to double-check. The prime factorization of 160 is $2^5 \times 5$, which can be rewritten as $2^4 \times 2 \times 5$. This gives us a perfect square of $2^4$ (which is 16). So, $\sqrt{160}$ simplifies to $\sqrt{16 \times 10}$, which is $4\sqrt{10}$. We already know that $4\sqrt{10}$ is not equivalent to $2\sqrt{10}$, so this option is out.

Option 5: $5\sqrt{8}$

This one requires a little more work. We need to simplify $5\sqrt{8}$ and see if it matches our simplified form of $\sqrt{40}$ ($2\sqrt{10}$). First, let's simplify $\sqrt{8}$. The prime factorization of 8 is $2 \times 2 \times 2$, which is $2^3$. We can rewrite this as $2^2 \times 2$, so $\sqrt{8}$ becomes $\sqrt{4 \times 2}$, which simplifies to $2\sqrt{2}$. Now, we have $5\sqrt{8} = 5 \times 2\sqrt{2} = 10\sqrt{2}$. Hmmm, this still doesn't look like $2\sqrt{10}$. However, a clever way to confirm is to square both numbers to see if they are the same: $(2\sqrt{10})^2 = 4 * 10 = 40$ and $(5\sqrt{8})^2 = 25 * 8 = 200$. Therefore, we can confidently say it is not equivalent to the original question.

Final Answer: The Equivalent Expressions

Alright, guys, we've cracked the case! After carefully analyzing each option, we've found the expressions that are equivalent to $\sqrt{40}$. The correct answers are:

• $2\sqrt{10}$
• $40^{\frac{1}{2}}$

We successfully simplified $\sqrt{40}$ and recognized its equivalent forms, even when they were disguised as exponents. Remember, practice makes perfect! The more you work with radicals and exponents, the easier it will become to spot these equivalencies. Keep up the great work, and you'll be a math whiz in no time!