Question

Math

Posted 5 months ago

```
Rewrite the expression in the form $y^{n}$.
Write the exponent as an integer, fraction, or an exact decimal (not a mixed number).
$\sqrt[4]{y^{\frac{7}{3}} y^{\frac{1}{3}}}=$
$\square$
```

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Answer from Sia

Posted 5 months ago

Solution by Steps

step 1

To simplify the given expression, we will use the properties of exponents

step 2

The expression inside the radical can be combined using the property $y^{a} \cdot y^{b} = y^{a+b}$

step 3

Adding the exponents $\frac{7}{3}$ and $\frac{1}{3}$ gives us $y^{\frac{7}{3} + \frac{1}{3}} = y^{\frac{8}{3}}$

step 4

The fourth root of $y^{\frac{8}{3}}$ can be written as $y^{\frac{8}{3} \cdot \frac{1}{4}}$

step 5

Multiplying the exponents gives us $y^{\frac{8}{3} \cdot \frac{1}{4}} = y^{\frac{8}{12}} = y^{\frac{2}{3}}$

[1] Answer

$y^{\frac{2}{3}}$

Key Concept

Properties of Exponents

Explanation

To simplify an expression involving radicals and exponents, we combine like bases using the property of exponents that states $y^{a} \cdot y^{b} = y^{a+b}$, and then apply the property that the nth root of $y^{m}$ is $y^{\frac{m}{n}}$.

Solution by Steps

step 1

To determine the sign of $f(x)$ on the interval $-\frac{1}{5}<x<2$, we examine the sign of each factor in the product

step 2

The factor $(5x+1)$ is positive for $x>-0.2$ since $-0.2$ is its zero and the leading coefficient is positive

step 3

The factor $(4x-8)$ is positive for $x>2$, but since our interval is $-\frac{1}{5}<x<2$, this factor is negative within the interval

step 4

The factor $(x+6)$ is positive for all $x>-6$, which includes our entire interval

step 5

Since two factors are positive and one is negative within the interval, their product $f(x)$ is negative for $-\frac{1}{5}<x<2$

[2] Answer

(B) $f$ is always negative on the interval.

Key Concept

Sign of a Polynomial Function on an Interval

Explanation

To determine the sign of a polynomial function on a given interval, we look at the signs of the individual factors of the function over that interval. If there is an odd number of negative factors, the function is negative on the interval.

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