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Simplify 161/4 x 2⁻³ x 30

Mathematics

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6 parts
2

C. 16\text{C. } 16

3

A. 4.5\text{A. } 4.5

4

A. 5.0%\text{A. } 5.0\%

5

D. 0.08\text{D. } 0.08

6

B. 1.00×102\text{B. } 1.00 \times 10^2

7

B. 10335\text{B. } 1033_5

Step-by-step solution

17 steps

Here are the solutions to the problems:

Part 2· 2 steps

Simplify 1614×2−3×3016^{\frac{1}{4}} \times 2^{-3} \times 3^0

  1. Evaluate each term in the expression.

    1614=164=216^{\frac{1}{4}} = \sqrt[4]{16} = 2 2−3=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8} 30=13^0 = 1
  2. Multiply the results.

    2×18×1=28=142 \times \frac{1}{8} \times 1 = \frac{2}{8} = \frac{1}{4}

    The calculated answer is 14\frac{1}{4}. However, this is not among the given options (A. 4, B. 8, C. 16, D. 32). Assuming there might be a typo in the question and 2−32^{-3} was intended to be 232^3, let's re-evaluate:

    1614×23×30=2×8×1=1616^{\frac{1}{4}} \times 2^3 \times 3^0 = 2 \times 8 \times 1 = 16

    Based on this assumption, the answer would be 16.

    The most likely intended answer from the options, assuming a common typo, is C.

    C. 16\boxed{\text{C. } 16}

Part 3

A. 4.5\text{A. } 4.5

If xx varies directly as the cube root of yy and x=3x=3 when y=64y=64; find xx when y=216y=216.

  1. Write the direct variation equation.

    If xx varies directly as the cube root of yy, then x=ky3x = k \sqrt[3]{y}, where kk is the constant of proportionality.

  2. Use the given values (x=3,y=64x=3, y=64) to find the constant kk.

    3=k6433 = k \sqrt[3]{64} 3=k×43 = k \times 4 k=34k = \frac{3}{4}
  3. Use the constant kk and the new value (y=216y=216) to find xx.

    x=342163x = \frac{3}{4} \sqrt[3]{216} x=34×6x = \frac{3}{4} \times 6 x=184x = \frac{18}{4} x=4.5x = 4.5

    The final answer is A. 4.5\boxed{\text{A. } 4.5}.

Part 4

A. 5.0%\text{A. } 5.0\%

Find the rate per annum at which a sum of N52,200.00 generates a simple interest of N6,525.00 in 2122\frac{1}{2} years.

  1. Identify the simple interest formula and given values.

    The simple interest formula is I=PRTI = PRT.
    Given:
    I=N6,525.00I = N6,525.00
    P=N52,200.00P = N52,200.00
    T=212 years=2.5 yearsT = 2\frac{1}{2} \text{ years} = 2.5 \text{ years}

  2. Substitute the values into the formula and solve for RR.

    6525=52200×R×2.56525 = 52200 \times R \times 2.5 6525=130500×R6525 = 130500 \times R R=6525130500R = \frac{6525}{130500}
  3. Calculate the value of RR and convert to a percentage.

    R=0.05R = 0.05

    To express as a percentage, multiply by 100:

    R=0.05×100%=5%R = 0.05 \times 100\% = 5\%

    The final answer is A. 5.0%\boxed{\text{A. } 5.0\%}.

Part 5

D. 0.08\text{D. } 0.08

Express 0.07552 to two decimal places.

  1. Identify the digit in the second decimal place.

    The number is 0.07552. The second decimal place is 7.

  2. Look at the digit immediately after the second decimal place.

    The digit after 7 is 5.

  3. Apply rounding rules.

    Since the digit (5) is 5 or greater, round up the second decimal place.
    Rounding 7 up gives 8.

    So, 0.07552 rounded to two decimal places is 0.08.

    The final answer is D. 0.08\boxed{\text{D. } 0.08}.

Part 6

B. 1.00×102\text{B. } 1.00 \times 10^2

Evaluate 1.44(0.12)2\frac{1.44}{(0.12)^2}, stating your answer in standard form.

  1. Calculate the denominator (0.12)2(0.12)^2.

    (0.12)2=0.12×0.12=0.0144(0.12)^2 = 0.12 \times 0.12 = 0.0144
  2. Perform the division.

    1.440.0144\frac{1.44}{0.0144}

    To simplify, multiply the numerator and denominator by 10000:

    1.44×100000.0144×10000=14400144=100\frac{1.44 \times 10000}{0.0144 \times 10000} = \frac{14400}{144} = 100
  3. Express the result in standard form (scientific notation).

    100=1×102100 = 1 \times 10^2

    The final answer is B. 1.00×102\boxed{\text{B. } 1.00 \times 10^2}.

Part 7

B. 10335\text{B. } 1033_5

Find the product of 23523_5 and 21521_5, leaving your answer in base 5.

  1. Convert the base 5 numbers to base 10.

    235=(2×51)+(3×50)=10+3=131023_5 = (2 \times 5^1) + (3 \times 5^0) = 10 + 3 = 13_{10} 215=(2×51)+(1×50)=10+1=111021_5 = (2 \times 5^1) + (1 \times 5^0) = 10 + 1 = 11_{10}
  2. Multiply the base 10 numbers.

    1310×1110=1431013_{10} \times 11_{10} = 143_{10}
  3. Convert the product 14310143_{10} back to base 5.

    Divide 143 by 5 repeatedly and record the remainders:
    143÷5=28143 \div 5 = 28 remainder 3
    28÷5=528 \div 5 = 5 remainder 3
    5÷5=15 \div 5 = 1 remainder 0
    1÷5=01 \div 5 = 0 remainder 1
    Reading the remainders from bottom to top gives 103351033_5.

    The final answer is B. 10335\boxed{\text{B. } 1033_5}.

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