II. Find the square root.

(T) $\sqrt{49}$

(O) $\sqrt{16}$

E$\sqrt{100}$

(N) $\sqrt{81}$

(R) $\sqrt{36}$

(O) $\sqrt{4}$

(H) $\sqrt{900}$

(I) $\sqrt{144}$

(F) $\sqrt{6,400}$

(G) $\sqrt{1}$

(K) $\sqrt{10,000}$

III. Simplify.

(O) $15^2$

(R) $\sqrt{225}$

(L) $\sqrt{121}$

A$\sqrt{36} + \sqrt{64}$

(H) $\sqrt{16} + \sqrt{9}$

(S) $\sqrt{16 + 9}$

(Y) $\sqrt{0.25}$

(W) $11^2$

(M) $\sqrt{36 + 64}$

(H) $\sqrt{0.81}$

D$25^2$

(W) $\sqrt{625}$

(M) $\sqrt{25} - \sqrt{9}$

(O) $\sqrt{25 - 9}$

(P) $\sqrt{0.01}$

Question

03-11-2026
Mathematics

Answered

II. Find the square root.

(T) $\sqrt{49}$

(O) $\sqrt{16}$

E$\sqrt{100}$

(N) $\sqrt{81}$

(R) $\sqrt{36}$

(O) $\sqrt{4}$

(H) $\sqrt{900}$

(I) $\sqrt{144}$

(F) $\sqrt{6,400}$

(G) $\sqrt{1}$

(K) $\sqrt{10,000}$

III. Simplify.

(O) $15^2$

(R) $\sqrt{225}$

(L) $\sqrt{121}$

A$\sqrt{36} + \sqrt{64}$

(H) $\sqrt{16} + \sqrt{9}$

(S) $\sqrt{16 + 9}$

(Y) $\sqrt{0.25}$

(W) $11^2$

(M) $\sqrt{36 + 64}$

(H) $\sqrt{0.81}$

D$25^2$

(W) $\sqrt{625}$

(M) $\sqrt{25} - \sqrt{9}$

(O) $\sqrt{25 - 9}$

(P) $\sqrt{0.01}$

Verified Answer :

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Section I (Square side lengths): (I) $\boxed{8}$, (M) $\boxed{10}$.
Section II (Square roots): (T) $\boxed{7}$, (O) $\boxed{4}$, (E) $\boxed{10}$, (N) $\boxed{9}$, (R) $\boxed{6}$, (O) $\boxed{2}$, (H) $\boxed{30}$, (I) $\boxed{12}$, (F) $\boxed{80}$, (G) $\boxed{1}$, (K) $\boxed{100}$.
Section III (Simplification): (O) $\boxed{225}$, (R) $\boxed{15}$, (L) $\boxed{11}$, (A) $\boxed{14}$, (H) $\boxed{7}$, (S) $\boxed{5}$, (Y) $\boxed{0.5}$, (W) $\boxed{121}$, (M) $\boxed{10}$, (H) $\boxed{0.9}$, (D) $\boxed{625}$, (W) $\boxed{25}$, (M) $\boxed{2}$, (O) $\boxed{4}$, (P) $\boxed{0.1}$.

Explanation

Analyzing Section I: Square Side Lengths

We are tasked with solving a multi-part puzzle involving calculations with square roots, side lengths of squares, and simplifying numerical expressions. We will address each section systematically, showing the reasoning for every calculation.

The first section involves finding the side length of a square given its area. The area of a square is calculated by the formula $A = s^2$, where $s$ is the side length. Therefore, the side length is found by taking the square root of the area: $s = \sqrt{A}$.

For (I) Area = 64, the side length is $s = \sqrt{64} = 8$.
For (M) Area = 100, the side length is $s = \sqrt{100} = 10$.

Analyzing Section II: Square Roots

The second section requires calculating the principal square root of various numbers. The square root of a number $x$, denoted $\sqrt{x}$, is the non-negative number which, when multiplied by itself, equals $x$.

(T) $\sqrt{49} = 7$
(O) $\sqrt{16} = 4$
(E) $\sqrt{100} = 10$
(N) $\sqrt{81} = 9$
(R) $\sqrt{36} = 6$
(O) $\sqrt{4} = 2$
(H) $\sqrt{900} = \sqrt{9 \times 100} = 3 \times 10 = 30$
(I) $\sqrt{144} = 12$
(F) $\sqrt{6400} = \sqrt{64 \times 100} = 8 \times 10 = 80$
(G) $\sqrt{1} = 1$
(K) $\sqrt{10000} = \sqrt{100^2} = 100$

Analyzing Section III: Simplifying Expressions

The third section involves simplifying more complex expressions using squares, roots, and arithmetic. We must follow the order of operations, specifically evaluating terms inside roots or powers first.

(O) $15^2 = 15 \times 15 = 225$
(R) $\sqrt{225} = 15$
(L) $\sqrt{121} = 11$
(A) $\sqrt{36} + \sqrt{64} = 6 + 8 = 14$
(H) $\sqrt{16} + \sqrt{9} = 4 + 3 = 7$
(S) $\sqrt{16 + 9} = \sqrt{25} = 5$
(Y) $\sqrt{0.25} = 0.5$
(W) $11^2 = 11 \times 11 = 121$
(M) $\sqrt{36 + 64} = \sqrt{100} = 10$
(H) $\sqrt{0.81} = 0.9$
(D) $25^2 = 25 \times 25 = 625$
(W) $\sqrt{625} = 25$
(M) $\sqrt{25} - \sqrt{9} = 5 - 3 = 2$
(O) $\sqrt{25 - 9} = \sqrt{16} = 4$
(P) $\sqrt{0.01} = 0.1$

Conclusion

Through systematically applying the definitions of square roots and powers to each exercise in the provided sections, we have determined the numerical value for every item. These values can now be used to complete the puzzle. The summary below provides a consolidated reference of all computed values.

Examples

Understanding square roots and powers is foundational for many real-world applications. For instance, if you are planning to build a square garden, calculating the side length from the total area helps you determine how much fencing is required. Similarly, in architecture and design, simplifying expressions with roots is used to calculate diagonal lengths (like the hypotenuse of a triangle using the Pythagorean theorem, $a^2 + b^2 = c^2$). Mastery of these operations is essential for precise construction and spatial planning.