Styling Archives Widget Dropdown Menu

Hi! I’ve been trying to style the default wordpress archives widget dropdown menu to no avail.

In /wp-includes/default-widgets.php, I’ve added an id and class. I.e.:
<select name=”archive-dropdown” id=”archive_dropdown” class=”archive_dropdown” onchange=’document.location.href=this.options[this.selectedIndex].value;’> <option value=””><?php echo esc_attr(__(‘Select Month’)); ?></option> <?php wp_get_archives(apply_filters(‘widget_archives_dropdown_args’, array(‘type’ => ‘monthly’, ‘format’ => ‘option’, ‘show_post_count’ => $c))); ?> </select>

And in my theme style.css, added:
#archive_dropdown,
.archive_dropdown {
font-family: inherit;
cursor: crosshair !important;
}

I’ve also tried styling using these”
.widget_archive {}
.widget_archive ul {}
.widget_archive ul li {}
.widget_archive ul li a {}
.widget_archive select {}
.widget_archive option {}
as seen here
but also to no avail.

Additional info: I am using Google Chrome browser.

Has anyone managed to successfully style their dropdown menus? What codes did you use/edit? Thank you so much for any advice!