Question

Evaluate each of the following arithmetic expressions by performing the operations in the appropriate order. Check your answers with a calculator. a. \(37-2 \cdot(18-2 \cdot 5)+1^{2}\) b. \(3^{5}+2 \cdot 10^{2}\) c. \(243 /\left(36-3^{3}\right)\) d. \((75-2 \cdot 15) / 9\) e. \(7 \cdot 2^{3}-9 \cdot 2+5\) f. \(2^{5} \cdot 5^{2}\) g. \(2^{3} \cdot 2^{5}\) h. \(5^{3}+2^{6}\) i. \(1350 \div 75 \cdot 5-15 \cdot 2\) j. \(\left(3^{2}-4\right)^{2}\)

Short answer

Question: Evaluate the following expressions: a. \(37-2 \cdot(18-2 \cdot 5)+1^2\) b. \(3^5 + 2 \cdot 10^2\) c. \(243 / \left(36 - 3^3 \right)\) d. \((75 - 2 \cdot 15) / 9\) e. \(7 \cdot 2^3 - 9 \cdot 2 + 5\) f. \(2^5 \cdot 5^2\) g. \(2^3 \cdot 2^5\) h. \(5^3 + 2^6\) i. \(1350 \div 75 \cdot 5 - 15 \cdot 2\) j. \(\left(3^2 - 4\right)^2\) Answer: a. 22 b. 443 c. 27 d. 5 e. 43 f. 800 g. 256 h. 189 i. 60 j. 25

Step-by-step solution

Calculate inside parentheses

Perform the operation inside the parentheses first: \(18 - 2 \cdot 5 = 18 - 10 = 8\)

Evaluate the whole expression

Now replace the parentheses with the result and perform the operations from left to right: \(37 - 2 \cdot 8 + 1^2 = 37 - 16 + 1 = 21 + 1 = 22\) b. \(3^5 + 2 \cdot 10^2\)

Evaluate exponents

Calculate the value of the exponents first: \(3^5 = 243\) and \(10^2 = 100\)

Evaluate the whole expression

Now replace the exponents with the results and perform the operations from left to right: \(243 + 2 \cdot 100 = 243 + 200 = 443\) c. \(243 / \left(36 - 3^3 \right)\)

Evaluate the exponent

Calculate the value of the exponent first: \(3^3 = 27\)

Calculate inside the parentheses

Perform the operation inside the parentheses: \(36 - 27 = 9\)

Evaluate the whole expression

Now replace the parentheses with the result and perform the operations from left to right: \(243 / 9 = 27\) d. \((75 - 2 \cdot 15) / 9\)

Calculate inside parentheses

Perform the operation inside the parentheses first: \(75 - 2 \cdot 15 = 75 - 30 = 45\)

Evaluate the whole expression

Now replace the parentheses with the result and perform the operations from left to right: \(45 / 9 = 5\) e. \(7 \cdot 2^3 - 9 \cdot 2 + 5\)

Evaluate the exponent

Calculate the value of the exponent first: \(2^3 = 8\)

Evaluate the whole expression

Now replace the exponent with the result and perform the operations from left to right: \(7 \cdot 8 - 9 \cdot 2 + 5 = 56 - 18 + 5 = 38 + 5 = 43\) f. \(2^5 \cdot 5^2\)

Evaluate exponents

Calculate the value of the exponents first: \(2^5 = 32\) and \(5^2 = 25\)

Evaluate the whole expression

Now replace the exponents with the results and perform the operations from left to right: \(32 \cdot 25 = 800\) g. \(2^3 \cdot 2^5\)

Evaluate exponents

Calculate the value of the exponents first: \(2^3 = 8\) and \(2^5 = 32\)

Evaluate the whole expression

Now replace the exponents with the results and perform the operations from left to right: \(8 \cdot 32 = 256\) h. \(5^3 + 2^6\)

Evaluate exponents

Calculate the value of the exponents first: \(5^3 = 125\) and \(2^6 = 64\)

Evaluate the whole expression

Now replace the exponents with the results and perform the operations from left to right: \(125 + 64 = 189\) i. \(1350 \div 75 \cdot 5 - 15 \cdot 2\)

Perform divisions and multiplications

Perform the divisions and multiplications from left to right: \(1350 \div 75 = 18\) and \(15 \cdot 2 = 30\)

Evaluate the whole expression

Now replace the divisions and multiplications with the results and perform the operations from left to right: \(18 \cdot 5 - 30 = 90 - 30 = 60\) j. \(\left(3^2 - 4\right)^2\)

Evaluate the exponent

Calculate the value of the exponent first: \(3^2 = 9\)

Calculate inside parentheses

Perform the operation inside the parentheses: \(9 - 4 = 5\)

Evaluate the whole expression

Now replace the parentheses with the result and perform the operations from left to right: \(5^2 = 25\)