Simplify the following: $\frac{-\left(-x^3 y^2 z^4\right)^5\left(-x^3 y^2 z^3\right)^3}{\left[\left(x^2 y z\right)^0\right]^4}$ = $\square$ $\sqrt{x}$
Answer (2)
Simplify the denominator: [( x 2 yz ) 0 ] 4 = 1 .
Simplify the first term in the numerator: − ( − x 3 y 2 z 4 ) 5 = x 15 y 10 z 20 .
Simplify the second term in the numerator: ( − x 3 y 2 z 3 ) 3 = − x 9 y 6 z 9 .
Multiply and divide to get − x 24 y 16 z 29 = − x 2 47 y 16 z 29 x , so the answer is − x 2 47 y 16 z 29 .
Explanation
Understanding the Problem We are given the expression [ ( x 2 yz ) 0 ] 4 − ( − x 3 y 2 z 4 ) 5 ( − x 3 y 2 z 3 ) 3 and we want to simplify it and express it in the form □ x .
Simplifying the Denominator First, let's simplify the denominator. Since any non-zero expression raised to the power of 0 is 1, we have ( x 2 yz ) 0 = 1 . Therefore, the denominator becomes [( x 2 yz ) 0 ] 4 = [ 1 ] 4 = 1 .
Simplifying the First Term in the Numerator Now, let's simplify the first term in the numerator: − ( − x 3 y 2 z 4 ) 5 = − ( − 1 ) 5 ( x 3 ) 5 ( y 2 ) 5 ( z 4 ) 5 = − ( − 1 ) x 3 × 5 y 2 × 5 z 4 × 5 = x 15 y 10 z 20 .
Simplifying the Second Term in the Numerator Next, let's simplify the second term in the numerator: ( − x 3 y 2 z 3 ) 3 = ( − 1 ) 3 ( x 3 ) 3 ( y 2 ) 3 ( z 3 ) 3 = − x 3 × 3 y 2 × 3 z 3 × 3 = − x 9 y 6 z 9 .
Multiplying the Simplified Numerator Terms Now, let's multiply the two simplified terms in the numerator: x 15 y 10 z 20 × ( − x 9 y 6 z 9 ) = − x 15 + 9 y 10 + 6 z 20 + 9 = − x 24 y 16 z 29 .
Dividing by the Denominator Now, we divide the numerator by the denominator: 1 − x 24 y 16 z 29 = − x 24 y 16 z 29 .
Expressing in the Required Form We want to express the simplified expression in the form □ x . To do this, we can rewrite x 24 as x 23.5 × x 0.5 = x 2 47 x . Therefore, − x 24 y 16 z 29 = − x 2 47 y 16 z 29 x .
Final Answer Thus, the expression □ is − x 2 47 y 16 z 29 .
Examples
Understanding how to simplify expressions with exponents is crucial in many fields, such as physics and engineering. For example, when calculating the energy of a photon, E = h f , where h is Planck's constant and f is the frequency, and frequency can be related to wavelength λ by f = c / λ where c is the speed of light. If you are given an expression involving powers of these variables, you need to simplify it to understand the relationship between energy, wavelength, and frequency. Simplifying such expressions helps in making accurate calculations and predictions in various scientific applications.
To simplify the expression, we need to calculate each part of the numerator and the denominator separately. After simplification, we find that the result is − x 2 47 y 16 z 29 x .
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