Chapter 8: Q. 78 (page 771)

In the following exercises, simplify using absolute value signs as needed.
$\sqrt{150 m^{9} n^{3}}$ .
$\sqrt[3]{81 p^{7} q^{8}}$ .
$\sqrt[4]{162 c^{11} d^{12}}$ .

Short Answer

Expert verified

The simplified expressions are,

$5 |m^{3} n| \sqrt{6 m^{3} n}$ .
$3 |p^{2} q^{2}| \sqrt[3]{3 p q^{2}}$ .
$3 |c^{2} d^{3}| \sqrt[4]{2 c^{3}}$ .

Step by step solution

Step 1 Given expressions are,

$\sqrt{150 m^{9} n^{3}}$ .
$\sqrt[3]{81 p^{7} q^{8}}$ .
$\sqrt[4]{162 c^{11} d^{12}}$ .

The objective is, we need to find the simplified expression for the given expression.

Part a Step 2 150m9n3.

The given equation is, $\sqrt{150 m^{9} n^{3}}$ .

The product property of the roots is,

$\sqrt[n]{a b} = \sqrt[n]{a} . \sqrt[n]{b}$ and $\sqrt[n]{a} . \sqrt[n]{b} = \sqrt[n]{a b}$

Also, We know that, $\sqrt[n]{a^{n}} = |a|$
localid="1644750695651" $\sqrt{150 m^{9} n^{3}} = \sqrt{25 m^{6} n^{2} . 6 m^{3} n} = \sqrt{{(5 m^{3} n)}^{2} . 6 m^{3} n}$

Now use the product property in the obtained expression.

localid="1644750804999" $\sqrt{150 m^{9} n^{3}} = \sqrt{{(5 m^{3} n)}^{2}} . \sqrt{6 m^{3} n} = 5 |m^{3} n| \sqrt{6 m^{3} n}$

Part b Stepsrc="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHhtbG5zOndycz0iaHR0cDovL3d3dy53aXJpcy5jb20veG1sL21hdGhtbC1leHRlbnNpb24iIGhlaWdodD0iMTkiIHdpZHRoPSIxMCIgd3JzOmJhc2VsaW5lPSIxNSI+PCEtLU1hdGhNTDogPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtbj4yPC9tbj48L21hdGg+LS0+PGRlZnM+PHN0eWxlIHR5cGU9InRleHQvY3NzIi8+PC9kZWZzPjx0ZXh0IGZvbnQtZmFtaWx5PSJBcmlhbCIgZm9udC1zaXplPSIxNiIgdGV4dC1hbmNob3I9Im1pZGRsZSIgeD0iNC41IiB5PSIxNSI+MjwvdGV4dD48L3N2Zz4=" localid="1645117013119" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/76312d1f-9b53-4891-b720-5edb563a8abc.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T155227Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=8c83d7de9142230714d26c31db9cd1bb5f05dc7c373ccb4bc5cfb0007c8a67b5" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/76312d1f-9b53-4891-b720-5edb563a8abc.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T155202Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=dda7add9dd1088fbfe635e949b9bc3887fb0cc8ad907080a00d1fee49a6a49c9" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/76312d1f-9b53-4891-b720-5edb563a8abc.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T140956Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=4a85eefd804f29b121893bc5bf48cf2697477bcea56a869e5c93d28c3e58fb18" 2src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHhtbG5zOndycz0iaHR0cDovL3d3dy53aXJpcy5jb20veG1sL21hdGhtbC1leHRlbnNpb24iIGhlaWdodD0iMjYiIHdpZHRoPSI3MyIgd3JzOmJhc2VsaW5lPSIyMiI%2BPCEtLU1hdGhNTDogPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtbz4mI3hBMDs8L21vPjxtcm9vdD48bXJvdz48bW4%2BODE8L21uPjxtc3VwPjxtaT5wPC9taT48bW4%2BNzwvbW4%2BPC9tc3VwPjxtc3VwPjxtaT5xPC9taT48bW4%2BODwvbW4%2BPC9tc3VwPjwvbXJvdz48bW4%2BMzwvbW4%2BPC9tcm9vdD48L21hdGg%2BLS0%2BPGRlZnM%2BPHN0eWxlIHR5cGU9InRleHQvY3NzIi8%2BPC9kZWZzPjxwb2x5bGluZSBmaWxsPSJub25lIiBwb2ludHM9IjEyLC0yMCAxMSwtMjAgNSwwIDIsLTgiIHN0cm9rZT0iIzAwMDAwMCIgc3Ryb2tlLWxpbmVjYXA9InNxdWFyZSIgc3Ryb2tlLXdpZHRoPSIxIiB0cmFuc2Zvcm09InRyYW5zbGF0ZSg0LjUsMjMuNSkiLz48cG9seWxpbmUgZmlsbD0ibm9uZSIgcG9pbnRzPSI1LDAgMiwtOCAwLC03IiBzdHJva2U9IiMwMDAwMDAiIHN0cm9rZS1saW5lY2FwPSJzcXVhcmUiIHN0cm9rZS13aWR0aD0iMSIgdHJhbnNmb3JtPSJ0cmFuc2xhdGUoNC41LDIzLjUpIi8%2BPGxpbmUgc3Ryb2tlPSIjMDAwMDAwIiBzdHJva2UtbGluZWNhcD0ic3F1YXJlIiBzdHJva2Utd2lkdGg9IjEiIHgxPSIxNi41IiB4Mj0iNzEuNSIgeTE9IjMuNSIgeTI9IjMuNSIvPjx0ZXh0IGZvbnQtZmFtaWx5PSJBcmlhbCIgZm9udC1zaXplPSIxMiIgdGV4dC1hbmNob3I9Im1pZGRsZSIgeD0iNy41IiB5PSIxMyI%2BMzwvdGV4dD48dGV4dCBmb250LWZhbWlseT0iQXJpYWwiIGZvbnQtc2l6ZT0iMTYiIHRleHQtYW5jaG9yPSJtaWRkbGUiIHg9IjI3LjUiIHk9IjIyIj44MTwvdGV4dD48dGV4dCBmb250LWZhbWlseT0iQXJpYWwiIGZvbnQtc2l6ZT0iMTYiIGZvbnQtc3R5bGU9Iml0YWxpYyIgdGV4dC1hbmNob3I9Im1pZGRsZSIgeD0iNDAuNSIgeT0iMjIiPnA8L3RleHQ%2BPHRleHQgZm9udC1mYW1pbHk9IkFyaWFsIiBmb250LXNpemU9IjEyIiB0ZXh0LWFuY2hvcj0ibWlkZGxlIiB4PSI0OS41IiB5PSIxNSI%2BNzwvdGV4dD48dGV4dCBmb250LWZhbWlseT0iQXJpYWwiIGZvbnQtc2l6ZT0iMTYiIGZvbnQtc3R5bGU9Iml0YWxpYyIgdGV4dC1hbmNob3I9Im1pZGRsZSIgeD0iNTcuNSIgeT0iMjIiPnE8L3RleHQ%2BPHRleHQgZm9udC1mYW1pbHk9IkFyaWFsIiBmb250LXNpemU9IjEyIiB0ZXh0LWFuY2hvcj0ibWlkZGxlIiB4PSI2Ni41IiB5PSIxNSI%2BODwvdGV4dD48L3N2Zz4%3D" localid="1645116847287" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/67c1fb64-ce93-4a07-be6f-12bf91b2c442.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T155227Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=10063c5e0acc57715c4c8511717146344a4151a6243c23d790582740738edae0" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/67c1fb64-ce93-4a07-be6f-12bf91b2c442.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T155202Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=b94ffbbb4983e5a2a73896113bf38d477db6feee3c34d1a73378206650554041" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/67c1fb64-ce93-4a07-be6f-12bf91b2c442.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T140956Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=2961a8f6ee47eac0d4c50a06773443760bdef4e482e4a26495d0f472b0f8b322" 81p7q83.

$\sqrt[3]{81 p^{7} q^{8}} = \sqrt[3]{27 p^{6} q^{6} . 3 p q^{2}}$

Now use the product property in the obtained expression.

$\sqrt[3]{81 p^{7} q^{8}} = \sqrt[3]{27 p^{6} q^{6}} . \sqrt[3]{3 p q^{2}} = 3 |p^{2} q^{2}| \sqrt[3]{3 p q^{2}}$

Part c Step src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHhtbG5zOndycz0iaHR0cDovL3d3dy53aXJpcy5jb20veG1sL21hdGhtbC1leHRlbnNpb24iIGhlaWdodD0iMTkiIHdpZHRoPSIxMCIgd3JzOmJhc2VsaW5lPSIxNSI+PCEtLU1hdGhNTDogPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtbj4yPC9tbj48L21hdGg+LS0+PGRlZnM+PHN0eWxlIHR5cGU9InRleHQvY3NzIi8+PC9kZWZzPjx0ZXh0IGZvbnQtZmFtaWx5PSJBcmlhbCIgZm9udC1zaXplPSIxNiIgdGV4dC1hbmNob3I9Im1pZGRsZSIgeD0iNC41IiB5PSIxNSI+MjwvdGV4dD48L3N2Zz4=" localid="1645117021909" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/83bffafe-1693-45e2-83ce-09f26f977be1.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T155227Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=bcf38a6dd463d04aef3d3b744a8fd5be05fe701e97c958677c802d8eea797073" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/83bffafe-1693-45e2-83ce-09f26f977be1.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T155202Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=aa3e9bd14281ce9b23f2f529cceb6e328c7f3345459c40bf161739309d8197a7" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/83bffafe-1693-45e2-83ce-09f26f977be1.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T140956Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=3510569d13bc48f23a6c4de663253c953e62aba3ca1e1ab80b29c87a962f9397" 2 src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHhtbG5zOndycz0iaHR0cDovL3d3dy53aXJpcy5jb20veG1sL21hdGhtbC1leHRlbnNpb24iIGhlaWdodD0iMjYiIHdpZHRoPSI5MSIgd3JzOmJhc2VsaW5lPSIyMiI%2BPCEtLU1hdGhNTDogPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtcm9vdD48bXJvdz48bW4%2BMTYyPC9tbj48bXN1cD48bWk%2BYzwvbWk%2BPG1uPjExPC9tbj48L21zdXA%2BPG1zdXA%2BPG1pPmQ8L21pPjxtbj4xMjwvbW4%2BPC9tc3VwPjwvbXJvdz48bW4%2BNDwvbW4%2BPC9tcm9vdD48L21hdGg%2BLS0%2BPGRlZnM%2BPHN0eWxlIHR5cGU9InRleHQvY3NzIi8%2BPC9kZWZzPjxwb2x5bGluZSBmaWxsPSJub25lIiBwb2ludHM9IjEyLC0yMCAxMSwtMjAgNSwwIDIsLTgiIHN0cm9rZT0iIzAwMDAwMCIgc3Ryb2tlLWxpbmVjYXA9InNxdWFyZSIgc3Ryb2tlLXdpZHRoPSIxIiB0cmFuc2Zvcm09InRyYW5zbGF0ZSgwLjUsMjMuNSkiLz48cG9seWxpbmUgZmlsbD0ibm9uZSIgcG9pbnRzPSI1LDAgMiwtOCAwLC03IiBzdHJva2U9IiMwMDAwMDAiIHN0cm9rZS1saW5lY2FwPSJzcXVhcmUiIHN0cm9rZS13aWR0aD0iMSIgdHJhbnNmb3JtPSJ0cmFuc2xhdGUoMC41LDIzLjUpIi8%2BPGxpbmUgc3Ryb2tlPSIjMDAwMDAwIiBzdHJva2UtbGluZWNhcD0ic3F1YXJlIiBzdHJva2Utd2lkdGg9IjEiIHgxPSIxMi41IiB4Mj0iODkuNSIgeTE9IjMuNSIgeTI9IjMuNSIvPjx0ZXh0IGZvbnQtZmFtaWx5PSJBcmlhbCIgZm9udC1zaXplPSIxMiIgdGV4dC1hbmNob3I9Im1pZGRsZSIgeD0iMy41IiB5PSIxMyI%2BNDwvdGV4dD48dGV4dCBmb250LWZhbWlseT0iQXJpYWwiIGZvbnQtc2l6ZT0iMTYiIHRleHQtYW5jaG9yPSJtaWRkbGUiIHg9IjI3LjUiIHk9IjIyIj4xNjI8L3RleHQ%2BPHRleHQgZm9udC1mYW1pbHk9IkFyaWFsIiBmb250LXNpemU9IjE2IiBmb250LXN0eWxlPSJpdGFsaWMiIHRleHQtYW5jaG9yPSJtaWRkbGUiIHg9IjQ1LjUiIHk9IjIyIj5jPC90ZXh0Pjx0ZXh0IGZvbnQtZmFtaWx5PSJBcmlhbCIgZm9udC1zaXplPSIxMiIgdGV4dC1hbmNob3I9Im1pZGRsZSIgeD0iNTYuNSIgeT0iMTUiPjExPC90ZXh0Pjx0ZXh0IGZvbnQtZmFtaWx5PSJBcmlhbCIgZm9udC1zaXplPSIxNiIgZm9udC1zdHlsZT0iaXRhbGljIiB0ZXh0LWFuY2hvcj0ibWlkZGxlIiB4PSI2OC41IiB5PSIyMiI%2BZDwvdGV4dD48dGV4dCBmb250LWZhbWlseT0iQXJpYWwiIGZvbnQtc2l6ZT0iMTIiIHRleHQtYW5jaG9yPSJtaWRkbGUiIHg9IjgwLjUiIHk9IjE1Ij4xMjwvdGV4dD48L3N2Zz4%3D" localid="1645116881501" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/f57aad2f-415a-491e-83cb-e6a4f07fe50a.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T155227Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=198085b594f831fcb077a4d0c24164826f229627a06349e782cddb4fb0971f62" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/f57aad2f-415a-491e-83cb-e6a4f07fe50a.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T155202Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=10cfc35691c8615e94da51e4db06812e255f326ce3fde471a0b0b7354cec5cf8" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/f57aad2f-415a-491e-83cb-e6a4f07fe50a.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T140956Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=cc881db6a716070cfc182f640b0f20f02515b02779b7a9dbcf772e10bab42ac9" 162c11d124

$\sqrt[4]{162 c^{11} d^{12}} = \sqrt[4]{81 c^{8} d^{12} . 2 c^{3}}$

Now use the product property rule.
$\sqrt[4]{162 c^{11} d^{12}} = \sqrt[4]{81 c^{8} d^{12}} . \sqrt[4]{2 c^{3}} = 3 |c^{2} d^{3}| \sqrt[4]{2 c^{3}}$

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In the following exercises, simplify using absolute value signs as needed.
$\sqrt{150 m^{9} n^{3}}$ .
$\sqrt[3]{81 p^{7} q^{8}}$ .
$\sqrt[4]{162 c^{11} d^{12}}$ .

Short Answer

Step by step solution

Step 1 Given expressions are,

Part a Step 2 150m9n3.

One App. One Place for Learning.

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In the following exercises, simplify using absolute value signs as needed.150m9n3.81p7q83.162c11d124.

Short Answer

Step by step solution

Step 1 Given expressions are,

Part a Step 2 150m9n3.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Theoretical and Mathematical Physics

Mechanics Maths

Applied Mathematics

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

In the following exercises, simplify using absolute value signs as needed.
$\sqrt{150 m^{9} n^{3}}$ .
$\sqrt[3]{81 p^{7} q^{8}}$ .
$\sqrt[4]{162 c^{11} d^{12}}$ .