Solve 6¹ + 1⁶ + √81 = □²: Finding the Perfect Square Result

Q: Why are there two answers when finding square roots?

When solving $ x^2 = 16 $, we need both positive and negative values because both $ 4^2 = 16 $ and $ (-4)^2 = 16 $. Remember: negative × negative = positive!

Q: How do I calculate 1 raised to the 6th power?

The number 1 raised to any power always equals 1! So $ 1^6 = 1 \times 1 \times 1 \times 1 \times 1 \times 1 = 1 $. This is a helpful shortcut to remember.

Q: What's the difference between √81 and ±√81?

The symbol $ \sqrt{81} $ means the principal (positive) square root, which is 9. But when solving $ x^2 = 81 $, we write $ x = ±\sqrt{81} = ±9 $ to show both solutions.

Q: Do I need to follow order of operations here?

Yes! First calculate $ 6^1 = 6 $, $ 1^6 = 1 $, and $ \sqrt{81} = 9 $, then add: 6 + 1 + 9 = 16. Exponents and roots come before addition.

Q: How can I check if 4 and -4 are both correct?

Substitute each value: $ 4^2 = 16 $ ✓ and $ (-4)^2 = 16 $ ✓. Both equal our calculated left side of 16, so both answers are correct!

Perfect Squares with Mixed Operations

Indicate the missing number:

$6^1+1^6+\sqrt{81}=\textcolor{red}{☐}^2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine the missing value

00:03 Let's break down and calculate the powers

00:06 1 to the power of any number is always equal to 1

00:11 Let's break down 81 to 9 squared

00:20 The square root of any number squared is always equal to the number itself

00:34 Extract the root in order to isolate the unknown

00:47 Let's break down 16 to 4 squared

00:54 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Indicate the missing number:

$6^1+1^6+\sqrt{81}=\textcolor{red}{☐}^2$

Step-by-step solution

Let's simplify the direct calculation of the left side of the equation:

$6^1+1^6+\sqrt{81}=\textcolor{red}{☐}^2 \\ 6+1+9=\textcolor{red}{☐}^2\\ 16=\textcolor{red}{☐}^2\\$ When we calculated the numerical value of the term with the exponent and the term with the root, and remembered that raising the number 1 to any power will always give the result 1,

Now let's examine the equation that we obtained. On the left side we have the number 16 and on the right side we have a number (unknown) raised to the second power,

So we ask the question: "What number do we need to square to get the number 16?"

And the answer to that is of course - the number 4,

Therefore:

$16=\textcolor{red}{4}^2$ However, since we're dealing with an even power (power of 2), we must also consider the negative possibility,

Meaning it also holds true that:

$16=\textcolor{red}{(-4)}^2$

Therefore, the correct answer is answer C.

Final Answer

$4,\hspace{4pt}-4$

Key Points to Remember

Essential concepts to master this topic

Order of Operations: Calculate exponents and square roots before adding terms
Technique: Simplify 6¹ + 1⁶ + √81 = 6 + 1 + 9 = 16
Check: Verify both 4² = 16 and (-4)² = 16 satisfy the equation ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative square root solution
Don't assume only positive numbers can be squared to get 16! This misses half the answer. When x² = 16, both positive and negative values work because (-4)² = (-4)×(-4) = 16. Always consider both positive and negative solutions for even powers.

Practice Quiz

Test your knowledge with interactive questions

$6+\sqrt{64}-4=$

FAQ

Everything you need to know about this question

Why are there two answers when finding square roots?

When solving $x^2 = 16$ , we need both positive and negative values because both $4^2 = 16$ and $(-4)^2 = 16$ . Remember: negative × negative = positive!

How do I calculate 1 raised to the 6th power?

The number 1 raised to any power always equals 1! So $1^6 = 1 \times 1 \times 1 \times 1 \times 1 \times 1 = 1$ . This is a helpful shortcut to remember.

What's the difference between √81 and ±√81?

The symbol $\sqrt{81}$ means the principal (positive) square root, which is 9. But when solving $x^2 = 81$ , we write $x = ±\sqrt{81} = ±9$ to show both solutions.

Do I need to follow order of operations here?

Yes! First calculate $6^1 = 6$ , $1^6 = 1$ , and $\sqrt{81} = 9$ , then add: 6 + 1 + 9 = 16. Exponents and roots come before addition.

How can I check if 4 and -4 are both correct?

Substitute each value: $4^2 = 16$ ✓ and $(-4)^2 = 16$ ✓. Both equal our calculated left side of 16, so both answers are correct!

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