What Is (√3)⁴? The Easy Square Root Exponent Trick

Can you solve (√3)⁴ without a calculator? Yes. This quick square-root trick shows why the answer is simply 9.

Expressions with square roots and exponents often look more complicated than they really are. In the video, the expression (√3)⁴ is rewritten as four square roots multiplied together. Once the matching pairs are recognized, the problem becomes very easy.

The Quick Answer: (√3)⁴ = 9

(√3)⁴ = 9

The fastest way to see the answer is to group the four identical square roots into pairs:

(√3)⁴ = (√3 × √3) × (√3 × √3)

Each pair equals 3 because:

√3 × √3 = 3

Therefore:

(√3 × √3) × (√3 × √3) = 3 × 3 = 9

Step-by-Step Method

Step 1: Expand the exponent

An exponent tells you how many times to multiply the base by itself. Therefore, the fourth power means four copies of √3:

(√3)⁴ = √3 × √3 × √3 × √3

Step 2: Pair the square roots

Place the four factors into two pairs:

(√3 × √3) × (√3 × √3)

Step 3: Simplify each pair

A number multiplied by its own square root gives the number inside the radical when the two factors are both √3:

√3 × √3 = 3

Step 4: Multiply the results

There are two pairs, and each pair equals 3:

3 × 3 = 9

So the final answer is:

(√3)⁴ = 9

The Even-Exponent Shortcut

There is an even faster way to solve this problem. Since the exponent 4 is even, split it into two squares:

(√3)⁴ = [(√3)²]²

Now simplify the inner square:

(√3)² = 3

That leaves:

3² = 9

This shortcut works because the square and square root cancel each other when the expression is defined appropriately.

A Helpful Rule to Remember

(√a)² = a

When the exponent is 4, you can use the rule twice:

(√a)⁴ = [(√a)²]² = a²

For a = 3:

(√3)⁴ = 3² = 9

More Examples

Once you understand the pattern, similar expressions can be solved quickly:

Expression	Shortcut	Answer
(√2)⁴	2²	4
(√5)⁴	5²	25
(√7)⁴	7²	49
(√10)⁴	10²	100
(√12)⁴	12²	144

For example:

(√5)⁴ = 5² = 25

(√10)⁴ = 10² = 100

Common Mistakes to Avoid

The most common mistake is to treat (√3)⁴ as √(3⁴) without understanding why the rewrite works. In this example, the result is the same because the square root and the even exponent can be simplified correctly:

(√3)⁴ = √(3⁴) = √81 = 9

Another mistake is to think that √3 equals 3. It does not. √3 is an irrational number approximately equal to 1.732. The important point is that multiplying √3 by √3 produces exactly 3:

√3 × √3 = 3

Do not round √3 too early. Keeping the radical symbol until the final simplification gives an exact answer.

Practice Problems

Try these problems without a calculator:

1. (√2)⁴
2. (√6)⁴
3. (√8)⁴
4. (√11)⁴
5. (√15)⁴

Answer Key

1. (√2)⁴ = 4
2. (√6)⁴ = 36
3. (√8)⁴ = 64
4. (√11)⁴ = 121
5. (√15)⁴ = 225

Final Takeaway

When you see a square root raised to the fourth power, look for pairs. Each pair of identical square roots becomes the number inside the radical:

(√3)⁴ = (√3 × √3) × (√3 × √3) = 3 × 3 = 9

This simple pattern can save time on algebra homework, quizzes, and standardized test practice. Save this Maths Tricks guide and use the same method whenever you see an even power applied to a square root.