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8/19/2020Sophia :: Welcome1/24MILESTONEScore22/25You passed this Milestone22 questions were answered correctly.3 questions were answered incorrectly.1Simplify the following radical expression.RATIONALETo simplify this expression, we can use the Product Property of Radicals to separate the expressioninto two radicals.The cube root of can be written as the cube root of times the cube root of .Next, we can write each radical expression using a fractional exponent in order to simplify. Theindex of the radical determines the denominator of the fractional exponent. The index here is , soeach expression underneath the radical will be raised to the power.x cubed1 third
8/19/2020Sophia :: Welcome2/24Now that we have changed our original expression from a radical to fractional exponents, we canevaluate and simplify the two expressions that are raised to the power.to the power of evaluates to 3 because 3 raised to the 3rd power is (). Tosimplify , wemultiply the two exponents together.times equals and is simply . The expression simplifies to .CONCEPTApplying the Properties of Radicals 2Evaluate the following expression using the properties of logarithms.1 third1 thirdopen parentheses x cubed close parentheses to the power of 1 third end exponent31 third1x to the power of 1xlog subscript b open parentheses y close parenthesesequals x
8/19/2020Sophia :: Welcome3/24it can be rewritten as . Let's applythis to the first term, .tells us that raised to some number, , equals .raised to the power of is , so is equal to . We canrepeat this process with the next term, .tells us that raised to some number, , equals .raised to the power of is , so is equal to . Repeatthis one more time for the last term, .tells us that raised to some number, , equals raised to the power of is , so is Substitute thecalculated in for the log expressions to evaluate.Once the values are substituted, evaluate to addition.The expression evaluates to 6.CONCEPTIntroduction to Logarithms 3Rationalize the denominator and simplify the following expression:b to the power of x equals yx11x2x
8/19/2020Sophia :: Welcome4/24RATIONALEWhen working with radicals, we want to avoid having a radical in the denominator. Thefirst step in the process is to rationalize the denominator. To do this, multiply thenumerator and denominator by the conjugate of the denominator. The conjugate is abinomial with the opposite sign between its term, or .