Solve: 4096 x = 8<\/p>\n\n\n\n
The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n We are given the equation: 4096x=84096^x = 8<\/p>\n\n\n\n To solve for xx, it\u2019s helpful to express both 4096 and 8 as powers of the same base.<\/p>\n\n\n\n We know:<\/p>\n\n\n\n So we can rewrite the equation using base 2: (212)x=23(2^{12})^x = 2^3<\/p>\n\n\n\n Using the exponent rule (am)n=am\u22c5n(a^m)^n = a^{m \\cdot n}, we get: 212x=232^{12x} = 2^3<\/p>\n\n\n\n Since the bases are the same, we can set the exponents equal to each other: 12x=312x = 3<\/p>\n\n\n\n x=312=14x = \\frac{3}{12} = \\frac{1}{4}<\/p>\n\n\n\n x=14x = \\frac{1}{4}<\/p>\n\n\n\n This problem involves solving an exponential equation, where the variable xx is in the exponent. The key to solving such equations is to express both sides using the same base.<\/p>\n\n\n\n First, observe the numbers. The number 4096 might seem large, but it\u2019s a power of 2. In fact: 212=40962^{12} = 4096<\/p>\n\n\n\n Similarly, 8 is a small and well-known power of 2: 23=82^3 = 8<\/p>\n\n\n\n By writing both sides of the equation with base 2, we get: (212)x=23(2^{12})^x = 2^3<\/p>\n\n\n\n We then use the rule of exponents that states: (am)n=am\u22c5n(a^m)^n = a^{m \\cdot n}. Applying that here simplifies the left-hand side: 212x=232^{12x} = 2^3<\/p>\n\n\n\n Now that both sides have the same base (2), the only way the equation can be true is if the exponents are equal: 12x=312x = 3<\/p>\n\n\n\n Solving for xx, we divide both sides by 12: x=312=14x = \\frac{3}{12} = \\frac{1}{4}<\/p>\n\n\n\n This means that raising 4096 to the power of 14\\frac{1}{4} equals 8. That checks out because taking the fourth root of 4096 gives 8.<\/p>\n\n\n\n So, the final answer is: x=14x = \\frac{1}{4}<\/p>\n","protected":false},"excerpt":{"rendered":" Solve: 4096 x = 8 The correct answer and explanation is: We are given the equation: 4096x=84096^x = 8 Step 1: Express both sides as powers of 2 To solve for xx, it\u2019s helpful to express both 4096 and 8 as powers of the same base. We know: So we can rewrite the equation using […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-24219","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/24219","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=24219"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/24219\/revisions"}],"predecessor-version":[{"id":24220,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/24219\/revisions\/24220"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=24219"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=24219"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=24219"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Express both sides as powers of 2<\/h3>\n\n\n\n
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Step 2: Apply the rule of exponents<\/h3>\n\n\n\n
Step 3: Solve for xx<\/h3>\n\n\n\n
\u2705 Final Answer:<\/h3>\n\n\n\n
\n\n\n\n\ud83d\udcd8 Explanation (300 words):<\/h3>\n\n\n\n