Question

question 1-13
use the dropdown menu to indicate the justification of the previous step to the current step when solving the equation:
$5 + 2d^{\frac{3}{2}} = 21$

statement	justification
2. $2d^{\frac{3}{2}} = 16$	2.
3. $d^{\frac{3}{2}} = 8$	3.
4. $d = 4$	4.

Answer:

To solve for the justifications, we analyze each step:

Step 2 Justification:

From \( 5 + 2d^{\frac{2}{3}} = 21 \) to \( 2d^{\frac{2}{3}} = 16 \), we subtract 5 from both sides of the equation. This is the Subtraction Property of Equality (subtracting the same number from both sides of an equation keeps it balanced).

Step 3 Justification:

From \( 2d^{\frac{2}{3}} = 16 \) to \( d^{\frac{2}{3}} = 8 \), we divide both sides of the equation by 2. This is the Division Property of Equality (dividing both sides of an equation by the same non - zero number keeps it balanced).

Step 4 Justification:

We have \( d^{\frac{2}{3}} = 8 \). First, we can rewrite \( d^{\frac{2}{3}} \) as \( (d^{\frac{1}{3}})^2 \). So, \( (d^{\frac{1}{3}})^2=8 \). Taking the square root of both sides, we get \( d^{\frac{1}{3}}=\pm\sqrt{8}=\pm2\sqrt{2} \)? Wait, no, let's do it correctly. Wait, \( 8 = 2^3 \). And \( d^{\frac{2}{3}}=8 \) can be rewritten using the property \( a^{\frac{m}{n}}=\sqrt[n]{a^m} \). So, \( \sqrt[3]{d^2}=8 \), then \( d^2 = 8^3=512 \)? No, that's not right. Wait, another approach: If we have \( d^{\frac{2}{3}} = 8 \), we can raise both sides to the power of \( \frac{3}{2} \) because \( (\frac{2}{3})\times(\frac{3}{2}) = 1 \). So, \( (d^{\frac{2}{3}})^{\frac{3}{2}}=8^{\frac{3}{2}} \). But \( 8^{\frac{3}{2}}=(\sqrt{8})^3=(2\sqrt{2})^3 = 16\sqrt{2} \)? No, that's wrong. Wait, maybe the original equation has a typo, or maybe \( d^{\frac{2}{3}} \) is \( d^{2/3} \) and we assume that we are dealing with real numbers and positive roots (since the final answer is \( d = 4 \)). Let's check with \( d = 4 \): \( 4^{\frac{2}{3}}=(4^{\frac{1}{3}})^2 \). \( 4^{\frac{1}{3}}=\sqrt[3]{4} \), \( (\sqrt[3]{4})^2=\sqrt[3]{16}\approx2.5198\), and \( 2\times2.5198 + 5\approx10.0396
eq21 \). Wait, maybe the exponent is \( \frac{3}{2} \) instead of \( \frac{2}{3} \)? Let's re - evaluate. If the equation was \( 5 + 2d^{\frac{3}{2}}=21 \):

• Step 2: \( 2d^{\frac{3}{2}}=16 \) (subtraction property)
• Step 3: \( d^{\frac{3}{2}} = 8 \) (division property)
• Step 4: Raise both sides to the \( \frac{2}{3} \) power: \( (d^{\frac{3}{2}})^{\frac{2}{3}}=8^{\frac{2}{3}} \). \( 8^{\frac{2}{3}}=(\sqrt[3]{8})^2=2^2 = 4 \), so \( d = 4 \). Ah, so probably the exponent is \( \frac{3}{2} \) (a typo in the original problem, maybe \( d^{\frac{3}{2}} \) instead of \( d^{\frac{2}{3}} \)). So, from \( d^{\frac{3}{2}}=8 \) to \( d = 4 \), we use the Power of a Power Property (or raising both sides to a reciprocal power to isolate the variable). Specifically, if we have \( d^{\frac{3}{2}}=8 \), and we raise both sides to the \( \frac{2}{3} \) power, \( (d^{\frac{3}{2}})^{\frac{2}{3}}=d^{(\frac{3}{2}\times\frac{2}{3})}=d^1=d \), and \( 8^{\frac{2}{3}}=(\sqrt[3]{8})^2 = 2^2=4 \). So the justification for step 4 is Power of a Power Property (or raising both sides to the \( \frac{2}{3} \) power, which is the inverse operation of the exponent \( \frac{3}{2} \) on \( d \)).

Final Justifications:

1. Subtraction Property of Equality
2. Division Property of Equality
3. Power of a Power Property (or raising both sides to the \( \frac{2}{3} \) power to solve for \( d \))

If we assume the exponent was a typo and it's \( d^{\frac{3}{2}} \) (to get \( d = 4 \) as the final answer), the justifications are as above. If we take the original exponent \( \frac{2}{3} \), there is a mistake in the final step (since \( d = 4 \) does not satisfy \( 5+2d^{\frac{2}{3}} = 21 \)). But based on the final answer \( d = 4 \), we can conclude that the exponent was likely \( \frac{3}{2} \) and the justifications are:

1. Subtraction Property of Equality
2. Division Property of Equality
3. Raising both sides to the \( \frac{2}{3} \) power (Power of a Power Property)