Which expression is equivalent to (16 x Superscript 8 Baseline y Superscript negative 12 Baseline) Superscript one-half?
Negative 4 x Superscript 4 Baseline y Superscript 6
Negative 8 x Superscript 4 Baseline y Superscript 6
StartFraction 4 x Superscript 4 Baseline Over y Superscript 6 EndFraction
StartFraction 8 x Superscript 4 Baseline Over y Superscript 6 Baseline EndFraction<\/p>\n\n\n\n
The correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n To solve the expression ( (16x^8 y^{-12})^{\\frac{1}{2}} ), we will apply the rules of exponents. Specifically, we will use the power of a product rule and the power of a quotient rule to simplify.<\/p>\n\n\n\n The expression that is equivalent to ( (16x^8 y^{-12})^{\\frac{1}{2}} ) is: Thus, the correct answer is: The problem involves simplifying the expression ( (16x^8 y^{-12})^{\\frac{1}{2}} ) using exponent rules. The primary rules at play are the power of a product rule ( (ab)^n = a^n b^n ), which allows us to apply the exponent to each factor inside the parentheses, and the power of a power rule ( (x^m)^n = x^{m \\cdot n} ).<\/p>\n\n\n\n First, we simplify the constant ( 16 ) by taking its square root, which gives ( 4 ). Then, we simplify the variables by multiplying their exponents by ( \\frac{1}{2} ). For ( x^8 ), multiplying the exponent by ( \\frac{1}{2} ) gives ( x^4 ). For ( y^{-12} ), multiplying the exponent by ( \\frac{1}{2} ) gives ( y^{-6} ), which simplifies further to ( \\frac{1}{y^6} ).<\/p>\n\n\n\n By combining the simplified terms, we obtain ( 4x^4 \\cdot \\frac{1}{y^6} ), or equivalently ( \\frac{4x^4}{y^6} ). This expression matches the third option, making it the correct answer.<\/p>\n\n\n\n Understanding these rules is key to handling algebraic expressions involving exponents, especially when dealing with fractional or negative powers.<\/p>\n","protected":false},"excerpt":{"rendered":" Which expression is equivalent to (16 x Superscript 8 Baseline y Superscript negative 12 Baseline) Superscript one-half?Negative 4 x Superscript 4 Baseline y Superscript 6Negative 8 x Superscript 4 Baseline y Superscript 6StartFraction 4 x Superscript 4 Baseline Over y Superscript 6 EndFractionStartFraction 8 x Superscript 4 Baseline Over y Superscript 6 Baseline EndFraction The […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-149922","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/149922","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=149922"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/149922\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=149922"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=149922"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=149922"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step-by-Step Breakdown:<\/h3>\n\n\n\n
\n
[
(16x^8 y^{-12})^{\\frac{1}{2}}
]
The first step is to apply the exponent ( \\frac{1}{2} ) to each term inside the parentheses separately.<\/li>\n\n\n\n
[
(16)^{\\frac{1}{2}} = \\sqrt{16} = 4
]
Since ( 16 ) is a perfect square, taking its square root results in ( 4 ).<\/li>\n\n\n\n\n
[
(x^8)^{\\frac{1}{2}} = x^{8 \\times \\frac{1}{2}} = x^4
]<\/li>\n\n\n\n
[
(y^{-12})^{\\frac{1}{2}} = y^{-12 \\times \\frac{1}{2}} = y^{-6}
]
This gives us ( y^{-6} ), which can also be written as ( \\frac{1}{y^6} ) to eliminate the negative exponent.<\/li>\n<\/ul>\n\n\n\n\n
After simplifying, the expression becomes:
[
4x^4 \\cdot y^{-6}
]
This is equivalent to:
[
\\frac{4x^4}{y^6}
]<\/li>\n<\/ol>\n\n\n\nFinal Answer:<\/h3>\n\n\n\n
[
\\frac{4x^4}{y^6}
]<\/p>\n\n\n\n
[
\\text{StartFraction 4 x Superscript 4 Baseline Over y Superscript 6 EndFraction}
]<\/p>\n\n\n\nExplanation (300 words):<\/h3>\n\n\n\n