Evaluate the expression √(3^4 + 12^2) by simplifying the powers and square root step by step.

Introduction / Context:

This arithmetic and algebra question checks your ability to simplify an expression that combines powers and a square root. Such expressions are often designed to look complicated at first but simplify to familiar integers. Recognising perfect squares inside a square root is a key skill in both school mathematics and competitive exams.

Given Data / Assumptions:

• The expression to evaluate is √(3^4 + 12^2).
• We use standard rules of exponents: a^n means a multiplied by itself n times.
• The square root √x refers to the principal (non negative) square root.
• All numbers involved are real and positive.

Concept / Approach:

The approach is straightforward: first evaluate the powers 3^4 and 12^2, then add the results, and finally take the square root of the sum. The key observation is that the final sum should ideally form a perfect square, allowing the square root to simplify to an integer. This is a classic pattern in exam questions.

Step-by-Step Solution:

Verification / Alternative check:

You can verify 15^2 quickly: 10^2 = 100, 5^2 = 25, and 2 * 10 * 5 = 100, so (10 + 5)^2 = 100 + 25 + 100 = 225. This confirms that 225 is indeed a perfect square and that 15 is the correct square root. Thus the calculation for the original expression is consistent.

Why Other Options Are Wrong:

The values 13, 17, and 19 squared are 169, 289, and 361 respectively, none of which equal 225. The value 9 squared is 81, which is much smaller. Therefore, none of these options can possibly be the square root of 225, which rules them out. Only 15 satisfies 15^2 = 225.

Common Pitfalls:

Some students accidentally compute 3^4 as 3 * 4 = 12 or 3^4 = 3^2 = 9, which is incorrect. Others might add 3^2 and 12^2 instead of 3^4 and 12^2. Carefully applying exponent rules and checking the multiplication steps avoids these errors.

Final Answer:

The value of √(3^4 + 12^2) is 15.