Question

working with radicals
complete the table below. each expression should be written in radical notation, written with rational exponents and evaluated using the calculator. the first one is done for you.

written in radical notation

written using rational exponents

evaluated to two decimal places

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$sqrt{2}$

$2^{\frac{1}{2}}$

1.41

$sqrt6{7^7}$

$7^{\frac{7}{6}}$

5.50 ×

$sqrt7{15^6}$

$15^{\frac{6}{7}}$ ✓

10.20 ×

$sqrt6{19^9}$ ×

$19^{\frac{2}{3}}$

82.82 ×

$sqrt9{21^8}$ ✓

$21^{\frac{8}{9}}$

15.06 ×

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Explanation:

Step1: Simplify $\sqrt[6]{7^7}$ and evaluate

First, rewrite $\sqrt[6]{7^7}=7^{\frac{7}{6}}=7^{1+\frac{1}{6}}=7\times7^{\frac{1}{6}}$.
Calculate $7^{\frac{1}{6}}\approx1.383$, then $7\times1.383\approx9.68$.

Step2: Simplify $\sqrt[7]{15^6}$ and evaluate

$\sqrt[7]{15^6}=15^{\frac{6}{7}}$.
Calculate $15^{\frac{6}{7}}=e^{\frac{6}{7}\ln(15)}$. $\ln(15)\approx2.708$, $\frac{6}{7}\times2.708\approx2.321$, $e^{2.321}\approx10.20$ (correction: precise calculation: $15^{6/7}=\sqrt[7]{15^6}=\sqrt[7]{11390625}\approx10.20$ was miscalculated, correct value: $15^{6/7}\approx10.20$ is wrong, actual: $15^6=11390625$, $\sqrt[7]{11390625}\approx10.20$ no, recalculate: $10^7=10000000$, $10.2^7=(10+0.2)^7\approx10^7+7*10^6*0.2+21*10^5*0.04+35*10^4*0.008+35*10^3*0.0016+21*10^2*0.00032+7*10*0.000064+0.2^7=10000000+1400000+84000+2800+56+0.672+0.000896=11486856.67$, which is larger than 11390625. So $10.15^7\approx(10+0.15)^7\approx10^7+7*10^6*0.15+21*10^5*0.0225+35*10^4*0.003375+35*10^3*0.00050625+21*10^2*0.0000759375+7*10*0.000011390625+0.15^7=10000000+1050000+47250+1181.25+17.71875+0.15946875+0.0007973125+1.70859375e-6\approx11508449.13$, still larger. $10.1^7\approx10^7+7*10^6*0.1+21*10^5*0.01+35*10^4*0.001+35*10^3*0.0001+21*10^2*0.00001+7*10*0.000001+0.1^7=10000000+700000+21000+350+3.5+0.021+0.00007+1e-7\approx10721353.52$. Now find between 10.1 and 10.15: $\frac{11390625-10721353.52}{11508449.13-10721353.52}\approx\frac{669271.48}{787095.61}\approx0.85$. So $10.1+0.05*0.85\approx10.14$, so $\approx10.14$)

Step3: Correct radical for $19^{\frac{9}{6}}$

Simplify $\frac{9}{6}=\frac{3}{2}$, so $19^{\frac{3}{2}}=\sqrt{19^3}=\sqrt{6859}$. Evaluate: $\sqrt{6859}\approx82.82$ was wrong, $19^3=6859$, $\sqrt{6859}=82.82$ is correct? No, $82^2=6724$, $83^2=6889$, so $\sqrt{6859}\approx82.82$ is correct, but radical form is $\sqrt{19^3}$ or $19\sqrt{19}$, simplified radical is $\sqrt{19^3}=19\sqrt{19}$.

Step4: Evaluate $\sqrt[9]{21^8}=21^{\frac{8}{9}}$

Calculate $21^{\frac{8}{9}}=e^{\frac{8}{9}\ln(21)}$. $\ln(21)\approx3.0445$, $\frac{8}{9}\times3.0445\approx2.7067$, $e^{2.7067}\approx15.06$ was wrong, recalculate: $15^9=38443359375$, $21^8=37822859361$. $\sqrt[9]{37822859361}\approx14.90$. Because $14.9^9=(15-0.1)^9\approx15^9-9*15^8*0.1+36*15^7*0.01-84*15^6*0.001+126*15^5*0.0001-126*15^4*0.00001+84*15^3*0.000001-36*15^2*0.0000001+9*15*0.000000001-0.1^9\approx38443359375-9*2562890625*0.1+36*170859375*0.01-84*11390625*0.001+126*759375*0.0001-126*50625*0.00001+84*3375*0.000001-36*225*0.0000001+9*15*1e-9-1e-9\approx38443359375-230660156.25+6150937.5-95681.25+9568.125-63.7875+0.2835-0.00081+0.000000135-1e-9\approx38443359375-230660156.25=38212699218.75+6150937.5=38218850156.25-95681.25=38218754475+9568.125=38218764043.125-63.7875=38218763979.3375+0.2835=38218763979.621-0.00081=38218763979.62019+0.000000135\approx38218763979.62$, which is larger than 37822859361. $14.8^9\approx(15-0.2)^9\approx15^9-9*15^8*0.2+36*15^7*0.04-84*15^6*0.008+126*15^5*0.0016-126*15^4*0.00032+84*15^3*0.000064-36*15^2*0.0000128+9*15*0.00000256-0.2^9\approx38443359375-9*2562890625*0.2+36*170859375*0.04-84*11390625*0.008+126*759375*0.0016-126*50625*0.00032+84*3375*0.000064-36*225*0.0000128+9*15*0.00000256-5.12e-7\approx38443359375-461320312.5+24603750-748992+15309-2003.25+1.728-0.010368+0.0003456-5.12e-7\approx38443359375-461320312.5=37982039062.5+24603750=38006642812.5-748992=38005893820.5+15309=38005909129.5-2003.25=38005907126.25+1.728=38005907127.978-0.010368=38005907127.9676+0.0003456\approx38005907127.968$. Now $\frac{37822859361-38005907127.968}{38218763979.62-38005907127.968}\approx\frac{-183047766.968}{212856851.652}\approx-0.86$. So $14.8+0.1*0.86\approx14.89$, so $\approx14.89$.

Answer:

Written in radical notation	Written using rational exponents	Evaluated to two decimal places
$\sqrt[6]{7^7}$	$7^{\frac{7}{6}}$	9.68
$\sqrt[7]{15^6}$	$15^{\frac{6}{7}}$	10.14
$\sqrt{19^3}$ (or $19\sqrt{19}$)	$19^{\frac{3}{2}}$ (simplified from $19^{\frac{9}{6}}$)	82.82
$\sqrt[9]{21^8}$	$21^{\frac{8}{9}}$	14.89