Question

1. inserting parentheses

insert one set of parentheses into each expression so that the expression has the given value.
expression\tvalue
10 ÷ 2 + 3 − 1\t1
4 + 5 · 8 − 11\t61
4 · 12 − 3 + 5 · 2\t32
5² − 6 + 2⁴ ÷ 4 · 2\t5

Explanation:

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

We want to get a value of \(1\). Let's try different groupings. If we group \(2 + 3\), we have \(10\div(2 + 3)-1\). Calculate inside the parentheses: \(2 + 3=5\). Then \(10\div5 = 2\), and \(2-1 = 1\). So the expression is \(10\div(2 + 3)-1\).

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

We want a value of \(61\). Let's group \(4 + 5\). So \((4 + 5)\cdot8-11\). Calculate inside the parentheses: \(4 + 5 = 9\). Then \(9\cdot8=72\), and \(72-11 = 61\). So the expression is \((4 + 5)\cdot8-11\).

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

We want a value of \(32\). Let's group \(12 - 3 + 5\). So \(4\cdot(12 - 3 + 5)\cdot2\)? Wait, no, let's check again. Wait, \(4\cdot(12-(3 + 5))\cdot2\)? No, wait the original expression is \(4\cdot12-3 + 5\cdot2\). Wait, maybe \(4\cdot(12 - 3)+5\cdot2\)? No, \(4\cdot9 + 10=36 + 10 = 46\). Wait, maybe \(4\cdot(12-(3 + 5\cdot2))\)? No, \(5\cdot2 = 10\), \(3 + 10 = 13\), \(12-13=-1\), \(4\cdot(-1)=-4\). Wait, maybe I made a mistake. Wait the value is \(32\). Let's try \(4\cdot(12 - 3 + 5)- 2\cdot0\)? No, the original expression has \(5\cdot2\). Wait, maybe \(4\cdot(12 - (3 + 5))\cdot2\) is wrong. Wait, let's re - express. The expression is \(4\times12-3 + 5\times2\). Let's group \(12-3 + 5\) as \(12-(3 + 5)=4\), then \(4\times4+5\times2=16 + 10 = 26\). No. Wait, maybe \(4\times(12 - 3)+5\times2\) is \(36 + 10 = 46\). Wait, maybe the expression is \(4\cdot12-(3 + 5\cdot2)\). \(5\cdot2 = 10\), \(3 + 10 = 13\), \(48-13 = 35\). No. Wait, maybe \(4\cdot(12 - 3 + 5)- 5\cdot2\)? No. Wait, perhaps I misread the expression. Wait the expression is \(4\cdot12-3 + 5\cdot2\), and the value is \(32\). Let's calculate \(4\times(12-(3 + 5))\times2\) is wrong. Wait, maybe the expression is \(4\cdot(12 - 3 + 5)- 2\times5\)? No. Wait, let's try \(4\cdot(12 - (3 + 5))+5\cdot2\). \(12-(3 + 5)=4\), \(4\times4 = 16\), \(5\times2 = 10\), \(16 + 10 = 26\). Hmm. Maybe the correct grouping is \(4\cdot(12 - 3)+5\cdot2- 4\)? No, the problem says insert one set of parentheses. Wait, maybe the expression is \(4\cdot(12-(3 + 5\cdot2))\) is wrong. Wait, maybe the expression is \(4\cdot12-(3 + 5)\cdot2\). \(3 + 5 = 8\), \(8\times2 = 16\), \(48-16 = 32\). Yes! So \(4\cdot12-(3 + 5)\cdot2\). Let's check: \(4\times12 = 48\), \((3 + 5)\times2=16\), \(48-16 = 32\). So the expression is \(4\cdot12-(3 + 5)\cdot2\).

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

We want a value of \(5\). Let's group \(5^{2}-(6 + 2^{4}\div4\cdot2)\). Calculate \(2^{4}=16\), \(16\div4 = 4\), \(4\times2 = 8\), \(6 + 8 = 14\), \(25-14 = 11\). No. Wait, group \(2^{4}\div(4\cdot2)\). So \(5^{2}-6 + 2^{4}\div(4\cdot2)\). \(2^{4}=16\), \(4\cdot2 = 8\), \(16\div8 = 2\), \(25-6 + 2=21\). No. Wait, group \(5^{2}-(6 + 2^{4})\div(4\cdot2)\). \(2^{4}=16\), \(6 + 16 = 22\), \(4\cdot2 = 8\), \(22\div8 = 2.75\), \(25-2.75 = 22.25\). No. Wait, group \((5^{2}-6)+2^{4}\div(4\cdot2)\). \(25-6 = 19\), \(16\div8 = 2\), \(19 + 2 = 21\). No. Wait, maybe \(5^{2}-(6 + 2^{4}\div4)\cdot2\). \(2^{4}\div4 = 4\), \(6 + 4 = 10\), \(10\times2 = 20\), \(25-20 = 5\). Yes! So the expression is \(5^{2}-(6 + 2^{4}\div4)\cdot2\).

Answer:

• For \(10\div2 + 3 - 1\) (value \(1\)): \(10\div(2 + 3)-1\)
• For \(4 + 5\cdot8 - 11\) (value \(61\)): \((4 + 5)\cdot8-11\)
• For \(4\cdot12 - 3 + 5\cdot2\) (value \(32\)): \(4\cdot12-(3 + 5)\cdot2\)
• For \(5^{2}-6 + 2^{4}\div4\cdot2\) (value \(5\)): \(5^{2}-(6 + 2^{4}\div4)\cdot2\)