Question

inserting parentheses

1. insert one set of parentheses into each expression so that the expression has the given value.

expression	value
$4 + 5 \cdot 8 - 11$	61
$4 \cdot 12 - 3 + 5 \cdot 2$	32
$5^2 - 6 + 2^4 \div 4 \cdot 2$	5

Explanation:

Step1: Analyze \(10\div2 + 3 - 1\) (Value = 1)

We want to make the result 1. Let's try grouping \(2 + 3 - 1\) first. So the expression becomes \(10\div(2 + 3 - 1)\). Calculate inside the parentheses: \(2+3 - 1=4\). Then \(10\div4 = 2.5\), not 1. Try grouping \(2 + 3\) as \(10\div(2 + 3)-1\). \(10\div5 - 1=2 - 1 = 1\). Wait, but the problem says one set of parentheses. Wait, \(10\div(2 + 3 - 1)\) was wrong. Wait, \(10\div(2 + 3)-1\) has two operations? No, the problem says one set. Wait, maybe \(10\div(2 + 3 - 1)\) was miscalculated. Wait \(2+3 - 1 = 4\), \(10\div4 = 2.5\). Wait, maybe \(10\div(2 + (3 - 1))\)? No, that's two sets. Wait, the correct way: \(10\div(2 + 3 - 1)\) is one set. Wait, no, maybe I made a mistake. Wait the value is 1. Let's see: \(10\div(2 + 3 - 1)\) gives \(10\div4 = 2.5\). Wait, maybe \( (10\div2)+3 - 1=5 + 3 - 1 = 7\). No. Wait, \(10\div(2 + 3 - 1)\) is not. Wait, maybe the first expression: \(10\div(2 + 3 - 1)\) is wrong. Wait, let's check the second expression: \(4 + 5\cdot8 - 11\) (Value = 61). We need to get 61. Let's see, \(4 + 5\cdot(8 - 11)\) is \(4 + 5\cdot(-3)=4 - 15=-11\). No. \( (4 + 5)\cdot8 - 11=9\cdot8 - 11 = 72 - 11 = 61\). Yes! So that's one set of parentheses: \((4 + 5)\cdot8 - 11\).

Step2: Analyze \(4 + 5\cdot8 - 11\) (Value = 61)

As above, \((4 + 5)\cdot8 - 11\). Calculate inside the parentheses: \(4 + 5 = 9\). Then \(9\cdot8 = 72\), \(72 - 11 = 61\). Correct.

Step3: Analyze \(4\cdot12 - 3 + 5\cdot2\) (Value = 32)

We need to get 32. Let's see, \(4\cdot(12 - 3 + 5)\cdot2\) is two sets. No, one set. \(4\cdot(12 - (3 + 5))\cdot2\) is two. Wait, \(4\cdot(12 - 3 + 5)\) would be \(4\cdot14 = 56\). No. Wait, \(4\cdot(12 - (3 + 5)\cdot2)\) no. Wait, \(4\cdot12 - (3 + 5)\cdot2=48 - 16 = 32\). Yes! So \(4\cdot12 - (3 + 5)\cdot2\). Wait, but the problem says one set of parentheses. Wait, \(4\cdot12 - (3 + 5)\cdot2\) has one set around \(3 + 5\). Let's check: \(4\cdot12 - (3 + 5)\cdot2 = 48 - 8\cdot2 = 48 - 16 = 32\). Correct.

Step4: Analyze \(5^{2}-6 + 2^{4}\div4\cdot2\) (Value = 5)

We need to get 5. Let's see, \(5^{2}-(6 + 2^{4}\div4\cdot2)\). Calculate inside: \(6 + 16\div4\cdot2=6 + 4\cdot2=6 + 8 = 14\). \(25 - 14 = 11\). No. \( (5^{2}-6 + 2^{4})\div(4\cdot2)\)? No, two sets. Wait, \(5^{2}-(6 + 2^{4}\div(4\cdot2))\). \(4\cdot2 = 8\), \(16\div8 = 2\), \(6 + 2 = 8\), \(25 - 8 = 17\). No. Wait, \( (5^{2}-(6 + 2^{4}))\div(4\cdot2)\)? No. Wait, the correct way: \(5^{2}-(6 + 2^{4}\div4)\cdot2\). \(16\div4 = 4\), \(6 + 4 = 10\), \(10\cdot2 = 20\), \(25 - 20 = 5\). Yes! So \(5^{2}-(6 + 2^{4}\div4)\cdot2\).

Answer:

1. For \(10\div2 + 3 - 1\) (Value = 1): \(10\div(2 + 3)-1\) (Wait, no, earlier mistake. Wait the correct one with one set: Wait, the problem says "one set of parentheses". So for \(10\div2 + 3 - 1\) to get 1: Let's re-express. \(10\div(2 + 3 - 1)\) is wrong. Wait, maybe \( (10\div(2 + 3))-1=2 - 1 = 1\). But that's one set: \(10\div(2 + 3)-1\) has parentheses around \(2 + 3\). Yes, one set. So \(10\div(2 + 3)-1\).

1. For \(4 + 5\cdot8 - 11\) (Value = 61): \((4 + 5)\cdot8 - 11\)

1. For \(4\cdot12 - 3 + 5\cdot2\) (Value = 32): \(4\cdot12 - (3 + 5)\cdot2\)

1. For \(5^{2}-6 + 2^{4}\div4\cdot2\) (Value = 5): \(5^{2}-(6 + 2^{4}\div4)\cdot2\)

(Note: The problem might have intended each expression to have one set of parentheses. The above are the correct parenthetical insertions to achieve the given values.)