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Simplify the Radical Expression: $\sqrt[5]{x}\\left(\\sqrt[5]{x^4}-4\\sqrt[5]{x^2}\\right)$
Mathematics
Grade 10 (Junior High School)
Question Content
Simplify the expression: $\sqrt[5]{x}\left(\sqrt[5]{x^4}-4\sqrt[5]{x^2}\\right)$
Correct Answer
$x - 4x^{\\frac{3}{5}}$ or $x - 4\\sqrt[5]{x^3}$
Detailed Solution Steps
1
Step 1: Rewrite the fifth roots as rational exponents using the rule $\sqrt[n]{a^m}=a^{\\frac{m}{n}}$. This transforms the expression to $x^{\\frac{1}{5}}\\left(x^{\\frac{4}{5}} - 4x^{\\frac{2}{5}}\\right)$.
2
Step 2: Apply the distributive property (distributive law of multiplication over subtraction) to multiply $x^{\\frac{1}{5}}$ by each term inside the parentheses: $x^{\\frac{1}{5}}\\cdot x^{\\frac{4}{5}} - x^{\\frac{1}{5}}\\cdot4x^{\\frac{2}{5}}$.
3
Step 3: Use the exponent rule $a^m \\cdot a^n = a^{m+n}$ to simplify each product: $x^{\\frac{1}{5}+\\frac{4}{5}} - 4x^{\\frac{1}{5}+\\frac{2}{5}}$.
4
Step 4: Calculate the sums of the exponents: $x^{1} - 4x^{\\frac{3}{5}}$, which can also be written in radical form as $x - 4\\sqrt[5]{x^3}$.
Knowledge Points Involved
1
Radical to Rational Exponent Conversion
The rule $\sqrt[n]{a^m}=a^{\\frac{m}{n}}$ allows converting radical expressions to rational exponents, making exponent operations easier. Here, $n$ is the index of the radical, $m$ is the exponent of the base inside the radical, and this applies when $a>0$ or when the expression is defined for non-positive $a$ with appropriate integer exponents.
2
Distributive Property of Multiplication
The distributive property states that $a(b - c) = ab - ac$, where $a$, $b$, and $c$ can be numbers, variables, or algebraic expressions. It is used to expand products of a single term and a polynomial expression.
3
Product Rule for Exponents
The product rule $a^m \\cdot a^n = a^{m+n}$ applies when multiplying two exponential terms with the same non-zero base $a$. It states that we add the exponents while keeping the base unchanged, which is only valid for real numbers $a$ and rational exponents $m,n$ where the expression is defined.
4
Rational Exponent to Radical Conversion
The reverse of the radical-to-exponent rule, $a^{\\frac{m}{n}}=\\sqrt[n]{a^m}$, allows converting simplified rational exponent expressions back to radical form for a more traditional radical representation, which is useful for presenting solutions in the required format.
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