Here are the solutions to the mathematics questions:

1. Simplify $27^{2/3}$

Question

Here are the solutions to the mathematics questions: 1. Simplify 272/3. 2. Express 7/19 as a percentage: correct to 1 decimal place. 3. The angle in a semicircle is . 4. Simplify rac10sqrt(32)

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the mathematics questions:

1. Simplify $27^{2/3}$

Rewrite the expression using roots and powers.
$27^{2/3} = (\sqrt[3]{27})^2$
Calculate the cube root of 27.
$\sqrt[3]{27} = 3$
Square the result.
$3^2 = 9$
The correct answer is (d) 9.
$\boxed{9}$
2. Express $7/19$ as a percentage: correct to 1 decimal place
Convert the fraction to a decimal.
$\frac{7}{19} \approx 0.368421$
Multiply by 100 to express as a percentage.
$0.368421 \times 100\% = 36.8421\%$
Round to one decimal place.
$36.8\%$
The correct answer is (d) 36.8%.
$\boxed{36.8\%}$
3. The angle in a semicircle is ____
The angle subtended by a diameter at any point on the circumference of a circle is a right angle.
The correct answer is (c) $90^\circ$ .
$\boxed{90^\circ}$
4. Simplify $\frac{10}{\sqrt{32}}$
Simplify the square root in the denominator.
$\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}$
Substitute the simplified square root back into the expression.
$\frac{10}{4\sqrt{2}}$
Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$ .
$\frac{10}{4\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{10\sqrt{2}}{4 \times 2} = \frac{10\sqrt{2}}{8}$
Simplify the fraction.
$\frac{10\sqrt{2}}{8} = \frac{5\sqrt{2}}{4}$
The correct answer is (a) $\frac{5\sqrt{2}}{4}$ .
$\boxed{\frac{5\sqrt{2}}{4}}$
5. If $y=7$ , then $dy/dx=$
The function is $y=7$ .
This is a constant function.
The derivative of any constant with respect to $x$ is 0.
$\frac{dy}{dx} = \frac{d}{dx}(7) = 0$
The correct answer is (c) 0.
$\boxed{0}$
6. The derivative of $x^n$ is ____
Using the power rule for differentiation, if $y = x^n$ , then $\frac{dy}{dx} = nx^{n-1}$ .
The correct answer is (c) $nx^{n-1}$ .
$\boxed{nx^{n-1}}$
7. Convert 1810 to base 3
Divide 1810 by 3 and record the remainder.
$1810 \div 3 = 603 \text{ remainder } 1$
Continue dividing the quotient by 3.
$603 \div 3 = 201 \text{ remainder } 0$ $201 \div 3 = 67 \text{ remainder } 0$ $67 \div 3 = 22 \text{ remainder } 1$ $22 \div 3 = 7 \text{ remainder } 1$ $7 \div 3 = 2 \text{ remainder } 1$ $2 \div 3 = 0 \text{ remainder } 2$
Read the remainders from bottom to top.
$1810_{10} = 2111001_3$
The options provided are 200, 201, 210, 220. None of these match the calculated answer. There might be a typo in the question or options. Assuming the question meant to ask for a smaller number or the options are incorrect. If we were to pick the closest looking option, it's not clear. However, based on the calculation, the answer is $2111001_3$ .
Let's re-check the options. It's possible the question meant to convert a different number or the options are for a different base. Given the options are small, it's highly likely the question is flawed or refers to a different problem. I will provide the correct conversion for 1810 to base 3.
$\boxed{2111001_3}$
8. The nth term of an A.P is ____
The formula for the nth term of an Arithmetic Progression (A.P.) is $a_n = a + (n-1)d$ , where $a$ is the first term and $d$ is the common difference.
The correct answer is (b) $a + (n-1)d$ .
$\boxed{a + (n-1)d}$
9. If $2x^5 = 14$ , Find $x$
Divide both sides by 2.
$x^5 = \frac{14}{2}$ $x^5 = 7$
Take the fifth root of both sides.
$x = \sqrt[5]{7}$
The options are 1, 2, 3, 4. None of these are $\sqrt[5]{7}$ . There might be a typo in the question. If the question was $2x = 14$ , then $x=7$ . If it was $x^2=14$ , $x=\sqrt{14}$ . If it was $2^x=14$ , $x=\log_2 14$ . Given the options, it's possible the question intended to be $2x=14$ . If we assume $2x=14$ , then $x=7$ . However, the question clearly states $2x^5=14$ .
Based on the given equation:
$\boxed{x = \sqrt[5]{7}}$
10. If $\sin \theta = 3/5$ , find $\cos \theta$
Use the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ .
$\left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1$ $\frac{9}{25} + \cos^2 \theta = 1$
Solve for $\cos^2 \theta$ .
$\cos^2 \theta = 1 - \frac{9}{25}$ $\cos^2 \theta = \frac{25}{25} - \frac{9}{25}$ $\cos^2 \theta = \frac{16}{25}$
Take the square root of both sides.
$\cos \theta = \pm\sqrt{\frac{16}{25}}$ $\cos \theta = \pm\frac{4}{5}$
Assuming $\theta$ is in the first quadrant (or where cosine is positive), $\cos \theta = 4/5$ .
The correct answer is (a) $4/5$ .
$\boxed{\frac{4}{5}}$
11. The distance between $(1,2)$ and $(4,6)$ is ____
Use the distance formula $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ .
Let $(x_1, y_1) = (1,2)$ and $(x_2, y_2) = (4,6)$ .
Substitute the coordinates into the formula.
$d = \sqrt{(4-1)^2 + (6-2)^2}$ $d = \sqrt{(3)^2 + (4)^2}$ $d = \sqrt{9 + 16}$ $d = \sqrt{25}$ $d = 5$
The correct answer is (c) 5.
$\boxed{5}$
12. Given $f(x) = x^2 - 4$ . Find $f(3)$ .
Substitute $x=3$ into the function $f(x)$ .
$f(3) = (3)^2 - 4$
Calculate the value.
$f(3) = 9 - 4$ $f(3) = 5$
The correct answer is (b) 5.
$\boxed{5}$
13. Evaluate $23 \times 54 \pmod{7}$
Find the remainder of 23 when divided by 7.
$23 = 3 \times 7 + 2$
So, $23 \equiv 2 \pmod{7}$ .
Find the remainder of 54 when divided by 7.
$54 = 7 \times 7 + 5$
So, $54 \equiv 5 \pmod{7}$ .
Multiply the remainders and find the remainder of the product when divided by 7.
$23 \times 54 \equiv 2 \times 5 \pmod{7}$ $23 \times 54 \equiv 10 \pmod{7}$
Find the remainder of 10 when divided by 7.
$10 = 1 \times 7 + 3$
So, $10 \equiv 3 \pmod{7}$ .
The correct answer is (b) 3.
$\boxed{3}$
14. Simplify $\frac{x-4}{4} - \frac{x-3}{6} - \frac{x-18}{12}$
Find the least common multiple (LCM) of the denominators 4, 6, and 12.
The LCM is 12.
Rewrite each fraction with the common denominator.
$\frac{x-4}{4} = \frac{3(x-4)}{12} = \frac{3x-12}{12}$ $\frac{x-3}{6} = \frac{2(x-3)}{12} = \frac{2x-6}{12}$ $\frac{x-18}{12} = \frac{x-18}{12}$
Combine the fractions.
$\frac{3x-12}{12} - \frac{2x-6}{12} - \frac{x-18}{12}$ $= \frac{(3x-12) - (2x-6) - (x-18)}{12}$
Distribute the negative signs and combine like terms in the numerator.
$= \frac{3x-12 - 2x+6 - x+18}{12}$ $= \frac{(3x-2x-x) + (-12+6+18)}{12}$ $= \frac{0x + 12}{12}$ $= \frac{12}{12}$ $= 1$
The options are $x-18$ , $x-6$ , $x-18$ , $x-6$ . None of these match the answer 1. There might be a typo in the question or options.
$\boxed{1}$
15. Three boys shared 10,500.00 in the ratio $6:7:8$ . Find the largest share.
Find the total number of ratio parts.
$6 + 7 + 8 = 21$
Calculate the value of one ratio part.
$\text{Value per part} = \frac{10,500}{21} = 500$
The largest share corresponds to the largest ratio part, which is 8.
$\text{Largest share} = 8 \times 500 = 4,000$
The correct answer is (a) 4,000.00.
$\boxed{4,000.00}$
16. If $43x = 16^{x+1}$ , Find the value of $x$
This question appears to have a typo. It is highly probable that it should be $4^{3x} = 16^{x+1}$ . I will solve it with this assumption.
Express both sides with the same base.
Since $16 = 4^2$ , we can write:
$4^{3x} = (4^2)^{x+1}$
Apply the exponent rule $(a^m)^n = a^{mn}$ .
$4^{3x} = 4^{2(x+1)}$ $4^{3x} = 4^{2x+2}$
Equate the exponents since the bases are the same.
$3x = 2x+2$
Solve for

Here are the solutions to the mathematics questions: 1. Simplify 272/3. 2. Express 7/19 as a percentage: correct to 1 decimal place. 3. The angle in a semicircle is . 4. Simplify rac10sqrt(32)

Answer

Step-by-step solution

Rewrite the expression using roots and powers.

Calculate the cube root of 27.

Square the result.

Convert the fraction to a decimal.

Multiply by 100 to express as a percentage.

Round to one decimal place.

Simplify the square root in the denominator.

Substitute the simplified square root back into the expression.

Rationalize the denominator by multiplying the numerator and denominator by 2\sqrt{2}2​.

Simplify the fraction.

The function is y=7y=7y=7.

The derivative of any constant with respect to xxx is 0.

Divide 1810 by 3 and record the remainder.

Continue dividing the quotient by 3.

Read the remainders from bottom to top.

Divide both sides by 2.

Take the fifth root of both sides.

Use the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1sin2θ+cos2θ=1.

Solve for cos⁡2θ\cos^2 \thetacos2θ.

Take the square root of both sides.

Use the distance formula d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}d=(x2​−x1​)2+(y2​−y1​)2​.

Substitute the coordinates into the formula.

Substitute x=3x=3x=3 into the function f(x)f(x)f(x).

Calculate the value.

Find the remainder of 23 when divided by 7.

Find the remainder of 54 when divided by 7.

Multiply the remainders and find the remainder of the product when divided by 7.

Find the remainder of 10 when divided by 7.

Find the least common multiple (LCM) of the denominators 4, 6, and 12.

Rewrite each fraction with the common denominator.

Combine the fractions.

Distribute the negative signs and combine like terms in the numerator.

Find the total number of ratio parts.

Calculate the value of one ratio part.

The largest share corresponds to the largest ratio part, which is 8.

Express both sides with the same base.

Apply the exponent rule (am)n=amn(a^m)^n = a^{mn}(am)n=amn.

Equate the exponents since the bases are the same.

Solve for

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More Mathematics Questions

Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$ .

The function is $y=7$ .

The derivative of any constant with respect to $x$ is 0.

Use the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ .

Solve for $\cos^2 \theta$ .

Use the distance formula $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ .

Substitute $x=3$ into the function $f(x)$ .

Apply the exponent rule $(a^m)^n = a^{mn}$ .