Example Question - fractional exponents
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Simplifying Complex Expression with Fractions and Exponents
To solve this expression, we should first simplify each part and then multiply them together. Let's start with the first part of the expression: The first part is a fraction with an exponent of -2: \(\left(\frac{{2p^3}}{{\sqrt[5]{q}}}\right)^{-2}\) Let's express the fifth root as fractional exponent and distribute the -2 exponent to each part inside the brackets: \(= \left(2p^3 \cdot q^{-\frac{1}{5}}\right)^{-2}\) \(= 2^{-2} \cdot \left(p^3\right)^{-2} \cdot \left(q^{-\frac{1}{5}}\right)^{-2}\) Now, apply the power of a power rule by multiplying the exponents: \(= 2^{-2} \cdot p^{-6} \cdot q^{\frac{2}{5}}\) The second part is: \(\frac{4}{p^{-1}q}\) Before multiplying both parts together, let's rewrite everything with positive exponents. For the first part, remember we need to move the terms with negative exponents to the denominator to make the exponents positive: \(= \frac{q^{\frac{2}{5}}}{2^2p^6}\) \(= \frac{q^{\frac{2}{5}}}{4p^6}\) For the second part: \(\frac{4}{p^{-1}q} = \frac{4p}{q}\), because moving \(p^{-1}\) to the numerator makes the exponent positive. Now, we multiply these two expressions together: \(\left(\frac{q^{\frac{2}{5}}}{4p^6}\right) \cdot \left(\frac{4p}{q}\right)\) Multiplying the numerators and the denominators separately: \(\frac{q^{\frac{2}{5}} \cdot 4p}{4p^6 \cdot q}\) Notice that \(4\) in the numerator and denominator cancels out and one p from the numerator cancels with one p from the denominator: \(\frac{q^{\frac{2}{5}} \cdot p}{p^6 \cdot q}\) Split up the exponents for the q terms and subtract the exponents for the p terms: \(= \frac{q^{\frac{2}{5}}}{p^{6-1}q}\) \(= \frac{q^{\frac{2}{5}}}{p^5q}\) Now, subtract the exponents for q, since dividing powers with the same base means you subtract the exponents: \(= \frac{1}{p^5q^{1-\frac{2}{5}}}\) \(= \frac{1}{p^5q^{\frac{5}{5}-\frac{2}{5}}}\) \(= \frac{1}{p^5q^{\frac{3}{5}}}\) And that's our result with positive indices: \[ \frac{1}{p^5q^{\frac{3}{5}}} \]
Solving Expression with Fraction Exponents in Vietnamese
Tôi sẽ giúp bạn giải câu 1(a) bằng tiếng Việt: Đề bài yêu cầu rút gọn biểu thức: \[ A = \left(\frac{2}{3}\right)^{-2} \cdot \left(\frac{-3}{2}\right)^{3} \cdot \left(\frac{1}{6}\right)^{0} \] Bây giờ, chúng ta sẽ tiến hành rút gọn từng phần của biểu thức: 1. \(\left(\frac{2}{3}\right)^{-2}\) là nghịch đảo của \(\left(\frac{2}{3}\right)^{2}\), vậy ta có: \[\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^{2} = \frac{3^2}{2^2} = \frac{9}{4}\] 2. \(\left(\frac{-3}{2}\right)^{3}\) là lập phương của \(\frac{-3}{2}\), nên ta có: \[\left(\frac{-3}{2}\right)^{3} = \left(\frac{-3}{2}\right) \cdot \left(\frac{-3}{2}\right) \cdot \left(\frac{-3}{2}\right) = \frac{-27}{8}\] 3. \(\left(\frac{1}{6}\right)^{0}\) mọi số mũ 0 đều bằng 1, nên ta có: \[\left(\frac{1}{6}\right)^{0} = 1\] Giờ, nhân tất cả các kết quả với nhau, ta có: \[A = \frac{9}{4} \cdot \frac{-27}{8} \cdot 1 = \frac{9 \cdot (-27)}{4 \cdot 8} = \frac{-243}{32}\] Vậy kết quả của biểu thức A là \(\frac{-243}{32}\). Hy vọng điều này giúp bạn giải quyết bài toán. Nếu bạn cần giúp đỡ với các phần khác của bài tập, đừng ngần ngại yêu cầu thêm giải đáp.
Solving a Negative Fractional Exponent Expression
The expression in the image is \((\frac{8}{27})^{-2/3}\). To solve this, we can apply the rule for negative exponents and fractional exponents. A negative exponent means that you take the reciprocal of the base, and a fractional exponent means you take the root of the base (the denominator of the fraction) and then raise it to the power of the numerator. Here's the step-by-step calculation: \[ \left(\frac{8}{27}\right)^{-2/3} = \left(\frac{27}{8}\right)^{2/3} \] Now we take the cube root of both the numerator and the denominator and then square them because the exponent is \(2/3\): \[ \left(\sqrt[3]{\frac{27}{8}}\right)^2 = \left(\frac{\sqrt[3]{27}}{\sqrt[3]{8}}\right)^2 = \left(\frac{3}{2}\right)^2 \] Finally, we square the result: \[ \left(\frac{3}{2}\right)^2 = \frac{3^2}{2^2} = \frac{9}{4} \] Therefore, \((\frac{8}{27})^{-2/3}\) equals \(\frac{9}{4}\) or 2.25.
Simplifying Fractional Exponents
To simplify the given expression \( \frac{2x^7y}{8x^9y^2} \), you'll need to divide the numerator by the denominator for each variable and the coefficients. Step 1 - Simplify the coefficients: Divide the coefficients \(2\) and \(8\): \[ \frac{2}{8} = \frac{1}{4} \] Step 2 - Simplify the x terms: When you divide terms with the same base, you subtract the exponents (according to the law of exponents): \[ x^7 / x^9 = x^{7-9} = x^{-2} \] Since a negative exponent indicates that you should take the reciprocal of the base raised to the positive exponent, \( x^{-2} \) can be rewritten as \( 1/x^2 \) if the expression is required to have only positive exponents. Step 3 - Simplify the y terms: For the y terms: \[ y / y^2 = y^{1-2} = y^{-1} \] Again, for positive exponents, \( y^{-1} \) would be \( 1/y \). Combining these results, the simplified expression is: \[ \frac{1}{4} \cdot \frac{1}{x^2} \cdot \frac{1}{y} \] Or more simply: \[ \frac{1}{4x^2y} \] This is your final simplified expression.
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