Evaluate the numerical expression `3^{-\frac{1}{3}}\cdot3^{\frac{7}{3}}`
3^(-1/3 + 7/3) = 3^2 = 9
To evaluate the numerical expression `3^{-\frac{1}{3}}\cdot3^{\frac{7}{3}}`, we'll start by simplifying each exponent separately.
First, let's simplify `3^{-\frac{1}{3}}`:
The negative exponent here means we need to take the reciprocal of the base raised to the positive power. Therefore, `3^{-\frac{1}{3}}` is equal to `1/(3^{\frac{1}{3}})`.
Next, let's simplify `3^{\frac{7}{3}}`:
To simplify this expression, we can rewrite the exponent `7/3` as `(3\cdot2 + 1)/3`.
Using the property of exponentiation `a^{m/n} = (a^m)^{1/n}`, we can rewrite `3^{\frac{7}{3}}` as `(3^2)^{\frac{1}{3}}\cdot3^{\frac{1}{3}}`.
Simplifying further, `(3^2)^{\frac{1}{3}}` is equal to `3^{2/3}`.
Now, let's substitute these simplified expressions back into the original expression:
`3^{-\frac{1}{3}}\cdot3^{\frac{7}{3}} = (1/(3^{\frac{1}{3}})) \cdot (3^{2/3})\cdot3^{\frac{1}{3}}`
Since `(1/(3^{\frac{1}{3}})) \cdot (3^{\frac{1}{3}})` is equivalent to 1 (the exponents cancel each other out), our expression simplifies to:
`3^{2/3}`
So, the numerical expression `3^{-\frac{1}{3}}\cdot3^{\frac{7}{3}}` is equal to `3^{2/3}`.
To evaluate the numerical expression `3^{-\frac{1}{3}}\cdot3^{\frac{7}{3}}`, we can simplify the expression first before performing the multiplication.
Let's start by evaluating the exponents separately:
1. For `3^{-\frac{1}{3}}`: The negative exponent indicates taking the reciprocal of the base. So, `3^{-\frac{1}{3}}` is equal to `1/(3^{\frac{1}{3}})`.
2. For `3^{\frac{7}{3}}`: We raise 3 to the power of `7/3`. This means taking the cube root of 3 raised to the power of 7.
Now, let's simplify each exponent:
1. `3^{-\frac{1}{3}}` becomes `1/(3^{\frac{1}{3}})`. The cube root of 3 is obtained by raising it to the power of 1/3. Thus, `1/(3^{\frac{1}{3}})` equals `1/(\sqrt[3]{3})`.
2. `3^{\frac{7}{3}}` can be simplified by finding the cube root of 3 raised to the power of 7. Therefore, `3^{\frac{7}{3}}` equals `(\sqrt[3]{3})^7`.
Now, we have `1/(\sqrt[3]{3})` multiplied by `(\sqrt[3]{3})^7`. Since the bases are the same, we can combine the expression:
`1/(\sqrt[3]{3})\cdot (\sqrt[3]{3})^7` is equivalent to `1\cdot (\sqrt[3]{3})^6` (subtracting the exponents).
Finally, we can simplify further:
`1\cdot (\sqrt[3]{3})^6` is equal to `(\sqrt[3]{3})^6`.
Therefore, the numerical expression `3^{-\frac{1}{3}}\cdot3^{\frac{7}{3}}` simplifies to `(\sqrt[3]{3})^6`.