If $$ a^{n+n} imes b^{m-n}=6^{4} $$ and $$ a^{n} imes b^{n}=16 $$ find The value of $$ a $$ and

Question

If $$ a^{n+n} 	imes b^{m-n}=6^{4} $$ and $$ a^{n} 	imes b^{n}=16 $$ find The value of $$ a $$ and

UpStudy AI · Accepted Answer

It appears that the problem statement is incomplete. You provided two equations involving the variables $$ a $$, $$ b $$, $$ m $$, and $$ n $$:

1. $$ a^{2n} \times b^{m-n} = 6^{4} $$
2. $$ a^{n} \times b^{n} = 16 $$

To find the values of $$ a $$ and $$ b $$, we typically need the values of $$ m $$ and $$ n $$, or additional equations that relate these variables. Without more information or constraints on $$ m $$ and $$ n $$, we cannot uniquely determine the values of $$ a $$ and $$ b $$.

**However, if we assume that the goal is to express $$ a $$ and $$ b $$ in terms of $$ m $$ and $$ n $$, here's how you might proceed:**

Let's start with the given equations:

1. $$ a^{2n} \times b^{m-n} = 6^{4} $$
2. $$ a^{n} \times b^{n} = 16 $$

**Step 1: Simplify the equations**

From the second equation:
$$
a^{n} \times b^{n} = 16 \implies (a \cdot b)^{n} = 16 \implies a \cdot b = 16^{\frac{1}{n}}
$$

**Step 2: Express one variable in terms of the other**

Assume $$ a = \frac{16^{\frac{1}{n}}}{b} $$.

**Step 3: Substitute into the first equation**

Substitute $$ a $$ into the first equation:
$$
\left(\frac{16^{\frac{1}{n}}}{b}\right)^{2n} \times b^{m-n} = 6^{4}
$$
$$
\frac{16^{2}}{b^{2n}} \times b^{m-n} = 6^{4}
$$
$$
256 \times b^{m - 3n} = 1296
$$
$$
b^{m - 3n} = \frac{1296}{256} = \frac{81}{16}
$$
$$
b = \left(\frac{81}{16}\right)^{\frac{1}{m - 3n}}
$$

**Step 4: Find $$ a $$ using the expression for $$ b $$**

$$
a = \frac{16^{\frac{1}{n}}}{\left(\frac{81}{16}\right)^{\frac{1}{m - 3n}}}
$$

**Conclusion:**

Without specific values for $$ m $$ and $$ n $$, $$ a $$ and $$ b $$ can only be expressed in terms of these variables as shown above. If additional information is provided (such as specific values for $$ m $$ and $$ n $$, or another equation relating these variables), a more concrete solution can be found.
Cannot determine the values of $$ a $$ and $$ b $$ with the given information.

If
and find
The value of and

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